A treatment of multiple scattering for thick and thin scatterers is derived using the sigma-formalism which is then used to give a statistical description of the Twiss functions with the scattering included. Full account is taken of the geometric correlation which occurs between the scattering angle and displacement of the scattered particle. The formalism is first derived for uncoupled beams, but is then extended to take coupling into account. Excellent agreement is demonstrated with Monte-Carlo data. The Twiss-Scatterer relations presented make it possible to include arbitrarily, thick scatterers as elements in accelerator codes

Maier, A T

Maier, A

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Thick scatterers seen through the twiss functions

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN | PS DIVISIONCERN/PS 98-061 (DI)THICK SCATTERERS SEEN THROUGH THE TWISSFUNCTIONSAndrew MaierAbstractA treatment of multiple scattering for thick and thin scatterers is de-rived using the -formalism which is then used to give a statistical descrip-tion of the Twiss functions with the scattering included. Full account istaken of the geometric correlation which occurs between the scatteringangle and displacement of the scattered particle. The formalism is rstderived for uncoupled beams, but is then extended to take coupling intoaccount. Excellent agreement is demonstrated with Monte-Carlo data.The Twiss-Scatterer relations presented make it possible to include arbit-rarily, thick scatterers as elements in accelerator codes.Geneva, Switzerland31st October1998A. Maier is currently a Ph. D student in the EP Division, CERN, working on the Proton-Ion Medical Machine Study (PIMMS) hosted by the PS Division, CERNIntroductionMultiple coulomb scattering can be an important source of emittance blowupand betatron mismatch when considering stripping foils, semi-destructive mon-itors, beam scrapers and specialist applications such as hadron therapy. Thethick-scatterer formulae derived in the present paper give a description of thebeam envelope and the emittances after scattering by a uniform, isotropic ab-sorber of arbitrary thickness. The derivation exploits the statistical descriptionof the beam given by the well-known -matrix and the nal expressions are writ-ten in terms of Twiss beam parameters. The modied Twiss parameters applyto the beam envelope and emittances and cannot be used for single particletracking, since the concept of betatron phase advance for an individual particleis lost when crossing the scatterer. The more usual thin-scatterer formulae areshown to be consistent with the thick scatterer results and the thick scattererformulae are shown to be in excellent agreement with Monte-Carlo tracking.Indications are also given for nding the energy loss and calculating emittancegrowth from the betatron mismatch caused by scattering.The -formalismThe -matrix formalism is briey reviewed following Buon [1], where the -matrix of the particles in a beam with transverse phase-space co-ordinates ~y isdened as: = h~yi~yT=*0BB@xx0zz01CCA(x x0z z0)+=0BB@hxxi hx0xi hzxi hz0xihxx0i hx0x0i hzx0i hz0x0ihxzi hx0zi hzzi hz0zihxz0i hx0z0i hzz0i hz0z0i1CCA(1)where the hyyi and hyy0i denotes the variance and the co-variance of the phase-space co-ordinates of the particle beam.A linear transformationM which transfers an initial vector ~y1, so that,~y2=M~y1has the following eect on the -matrix:2= h~y2~yT2i = hM~y1(M~y1)Ti =M h~y1~yT1i| {z }1MTand reveals that the -matrix transfers via2=M1MT: (2)1Thus, the knowledge of the -matrix at one point in a lattice is sucient tocalculate the -matrix at any other point, providedM is known.For an uncoupled beam, the co-variance terms between the planes disappear, sothatuc=0BB@hxxi hx0xi 0 0hxx0i hx0x0i 0 00 0 hzzi hz0zi0 0 hzz0i hz0z0i1CCA:It is possible to link the elements of the -matrix with the Courant and Snyderinvariants. Using the denition of the statistical emittancey,Ex= phx2ihx02i   hxx0i2; Ez= phz2ihz02i   hzz0i2(3)where  and  have their usual meanings as Twiss parameters and  = (1 + 2)=,uccan be written asuc=0BB@hxxi hx0xi 0 0hxx0i hx0x0i 0 00 0 hzzi hzz0i0 0 hz0zi hz0z0i1CCA=10BB@Exx Exx0 0 ExxExx0 00 0 Ezz Ezz0 0  EzzEzz1CCA:(4)Thick{scatterer transformations for an uncoupledbeamConsider a particle with co-ordinates ~y0= (y0; y00) at the entry to a scatterer (seeFigure 1). Multiple scattering will cause the particle to deviate from its originalpath. Instead of exiting the scatterer of thickness L at co-ordinates (y1; y01) thepath will be perturbed and the particle will exit at the new co-ordinates (y2; y02).Compared to the unperturbed path the particle will have received an additionalangular deection sand will have undergone a displacement y.It is intuitively obvious that there is a correlation between sand y, since aparticle which has received a large kick has also a higher probability of exitingthe absorber with a larger displacement. This correlation is entirely geometricin origin [2] and can be calculated to be,hsyi =L22s: (5)yEmittance is here dened as an area in phase space, i. e.  is included inside Exand Ez2Single particle(y0; y00)(y2; y02)(y1; y01)Uniform{thickness,isotropic scattererBeam(0; 0; 0; E0)Beam with no scattering(1; 1; 1; E1)Scattered beam(2; 2; 2; E2)LFigure 1: Eect of a thick scattererFurthermore, the variance of the displacement y can be calculated to behy2i =Lp302; (6)where 0=ph2si, the characteristic scattering angle, which can be calculatedfrom Highland's equation (19) as described in the Appendix A.Three relationships for the statistical eect of scattering can now be established.Firstly, the change in divergence,y02= y01+ s: (7)By squaring (7) and then averaging over the whole beam, the increase in diver-gence can be related to the characteristic scattering angle 0.hy022i = hy012i+ 2hy01si+ h2siwhere 2hy01si = 0 because y01and sare uncorrelated since scattering is iso-tropic. Thus the new divergence is given by,hy022i = hy021i+ 20: (8)3Similarly with the use of (6),hy22i = hy21i+ 2hy1yi+ hy2iwhere hy1; yi = 0 andhy22i = hy21i+Lp302: (9)Finally, the change in the cross-term yy0is given by,hy2y02i = h(y1+ y)(y01+ s)iwhich can be evaluated to behy2y02i = hy1y01i+ hsyi+ hy1si+ hy01yi: (10)Since scattering is isotropic and has no correlations with y1or y01, the last twoterms in (10) are zero. The second term, however, makes a nite contribution,since there is a correlation between the displacement y and the scatteringangle s, which is described by (5). Inserting (5) into (10) therefore leads tohy2y02i = hy1y01i+L220: (11)The changes due to scattering are described statistically in equations (8), (9)and (11) and provide the three relationships needed to solve the Twiss functionsafter scattering. Thus, using the correspondences in the matrix equation (4),E22= E11+ 20E22= E11+ Lp302 E22=  E11+ L220:(12)The right-hand side of each equation in (12) is fully evaluated by conditions atthe exit to the scatterer without scattering, and by the characteristics of thescatterer. Re-writing (12) with three constantsE22= AE22= BE22= C4and substituting into the Twiss relation below leads to the solution of the threeequations by1 + 2=  so that1 +CE22BE2=AE2and E2=pAB   C2: (13)Consequently E2can be evaluated asE2= E1s1+ 20E11+ L2203E1 1  L2E1202: (14)Backsubstitution into (12) now yields the modied Twiss functions after thescatterer.Extension to a coupled beamThe approach given in the previous section can now be generalised to coupledbeams. The -matrix for a coupled beam contains additional elements:coupled=0BB@hxxi hx0xi hzxi hz0xihxx0i hx0x0i hzx0i hz0x0ihxzi hx0zi hzzi hz0zihxz0i hx0z0i hzz0i hz0z0i1CCA: (15)By using the same technique as for the uncoupled beam | adding the scatteringand then squaring and averaging over the beam | the overall eect of scatteringcan be evaluated. Since scattering is isotropic, the elements of the -matrixwhich do not describe the coupling remain the same. The only additional workto be done is to deal with the three coupled terms in the matrix.Starting withhx2z2i = h(x1+ x)(z1+ z)iwhich can be transformed tohx2z2i = hx1z1i+ hx1zi+ hz1xi+ hxzi= hx1z1iresulting in no extra change, since scattering does not introduce any couplingbetween the x and z planes and all coupling terms cancel.Similarly,5hx2z02i = h(x1+ x)(z01+ zs)i= hx1z01i+ hx1zsi+ hz01xi+ hzsxi= hx1z01i:Finally the last term,hx02z02i = h(x01+ xs)(z01+ zsi= hx01z01i+ hx01zsi+ hz01xsi+ hxszsi= hx01z01i:All coupled terms are unchanged after scattering and only the uncoupled ele-ments change. Thus, it is possible to treat a coupled beam passing through ascatterer in the same way as an uncoupled beam, which is a welcome simplic-ation, although at rst, a little surprising.Approximation for a thin scattererIn practical applications it is often necessary to deal with thin scatterers, suchas stripping foils. For such an application, it is possible to make a thin scat-terer approximation. In a thin-scatterer, it is assumed, that the position of theparticle remains unchanged when passing through the scatterer, and only thechange in divergence has to be taken into account. Thus neglecting the termsdepending on L in (12) leads to,E22= E11+ 20E22= E11E22= E11(16)Equation (14) is still valid and yields,E2= E1s1 + 120E1: (17)Be sure that 120=E1 1 if the square root is to be expanded.Emittance dilution from betatron mismatchAs an application of the Twiss-Scatterer relations, consider the following ex-ample. For a luminescent screen introduced into a transfer line, the multiplescattering in the monitor will cause a betatron mismatch. If the beam then6enters a ring, the mismatch will cause lamentation and hence an increase inemittance according toEdiluted=12"sn+s  nsn2ns+ns#Es; (18)which can be found in [3]. Here the index s denotes the Twiss parameters afterscattering, while the index n denotes the non-scattered values, i. e. the valuesexpected at the entry to the ring for the beam without beam monitors.By isolating 2and 2by dividing by E2in equation (12) and inserting the newscattered values into equation (18) using the matched values for the line for 1and 1and using equation (14) to obtain the intermediate emittance blown-upby scattering (Esin (18)), it is possible to evaluate the emittance dilution ofthe mismatched beam due to scattering.Comparison with a Monte-Carlo simulationInput beam Beam after scatteringParametersused togeneratedistributionStatisticalestimationfromgenerateddistributionCalculatedbyTwiss-scatterrelationEstimatedstatisticallyfromtrackingresultsERMS;x2.30 2.30 77.3 76.5[ mm mrad]ERMS;z2.30 2.30 77.3 76.6[ mm mrad]x[m] 6.25 6.27 0.19 0.19z[m] 6.25 6.25 0.19 0.19x[m] 0.00 0.00 0.00 0.00z[m] 0.00 0.00 0.00 0.00Table 1: Comparison of calculated beam parameters and those obtained by trackingA simple comparison with tracking shows the validity of (12). Using a codefor designing passive spreading systems in hadron therapy [4], an uncorrelatedGaussian beam of 100 000 protons at 180 MeV with a momentum spread of0.1% and a spatial cut-o at 2 was generated and tracked through a copperfoil of 3.67 mm thickness. The absorber adds an average angle 0= 20:4 mradusing Highland's equation (19). The numerical data is collected in Table 1.The estimated Twiss parameters of the scattered beam obtained by statisticalanalysis of the distribution calculated by tracking agree extremely well withthose calculated using the Twiss-Scatterer relations of equation (12).7Appendix A. Multiple scatteringMultiple coulomb scattering of charged heavy particles is probably best de-scribed by the theory of Moliere [5] which has been extended and corrected byBethe [6], Fano [7] and Scott [8]. This theory describes the width and distribu-tion of the projected scattering angle, which is not quite Gaussian. However,Moliere's theory is somewhat complicated and dicult to apply, it is fortunatethat it is often possible to use a Gaussian approximation of the scattering distri-bution. Highland [9, 10] has developed such an approximation and he calculatesthe width by an astute observation that certain parameters in Moliere's theoryoccur in the same way as in the equation of the radiation length:0=14:1[MeV]pc[MeV]ZincrLLR1 +19log10LLR: (19)This equation is said to be accurate to 5% in the range 10 3 L=LR 100,except for very light elements, where the accuracy is supposed to be better than11% [11]. Note however, that there are a variety of similar equations quoted inthe literature, which all deviate at the one to two percent level. Especially forlow particle energies, such as those used in hadron therapy, the validity of (19)has to be questioned and it may be necessary to modify Highland's equation.The Highland formula is a t that takes into account the energy loss of thebeam. Thus equations (12) and (14) give the Twiss functions and emittance atthe new energy of the beam. To nd the energy loss the well known Bethe-BlochFormula can be used which can be found in [11]. It should be remembered thatif the energy loss is not accounted for by a reduction in the focusing strength ofthe line, additional betatron mismatch will be created.The formulation of scattering in terms of Twiss functions is, however, independ-ent of the method chosen to calculate the characteristic scattering angle. It issucient to be aware that Highland's formula is an approximation and shouldbe replaced by measured data when possible.Conclusions An analytically rigorous approach has been developed for evaluating scat-tering in the context of the Twiss functions. In particular, full account is taken for the geometrical correlation occur-ring between the scattering angle and the displacement of the scatteredparticle. The resultant Twiss-Scatterer relations allow the Twiss functions to becalculated in a continuous manner through a lattice, which opens the wayto incorporating scatterers into lattice programs and even using them formatching beam envelopes and emittances. This has already been imple-mented in WINAGILE[12].8 Excellent agreement with the Monte-Carlo data is achieved and demon-strates the accuracy achievable.AcknowledgementsThis work was performed at CERN, within the framework of the Proton-IonMedical Machine Study. The author would like to thank the PS Division forits hospitality in hosting this study and to acknowledge all the friendly helpand advice given by PS sta members. In particular he would like to thankP. Bryant for the continuous help in the study of the subject and for checkingthe calculations.References[1] J. Buon. A statistical description of a partice beam subjected to a linearcoupled betatron motion. Technical report, LAL/RT, April 1996.[2] R. K. Bock, K. Bos, S. Brandt, J. Myrheim, and M. Regler, editors. Formu-lae and Methods in Experimental Data Evaluation. The European PhysicalSociety, 1983.[3] P. J. Bryant. Beam transfer lines. In Turner [13], pages 219{238.[4] A. Maier. PhD thesis, University of Vienna, 1998.[5] G. Moliere. Theorie der Streuung schneller geladener Teilchen II. Zeitschriftfur Naturforschung, 3a:78{97, 1948.[6] H. A. Bethe. Moliere's theory of multiple scattering. Physical Review,89(6):1256{1266, 1952.[7] U. Fano. Inelastic collisions and the moliere theory of multiple scattering.Physical Review, 93(1):117{120, 1953.[8] W. T. Scott. The theory of small-angle multiple scattering of fast chargedparticles. Review of Modern Physics, 35:231{313, 1963.[9] V. L. Highland. Some practical remarks on multile scattering. NuclearInstruments and Methods, 129:497{499, 1975.[10] V. L. Highland. Erratum. Nuclear Instruments and Methods, 161:171, 1979.[11] C. Caso et al. Review of particle properties. The European Physical Journal,C3:1, 1998.[12] P. Bryant. Private communication. WINAGILE { lattice calculation program,1998.[13] S. Turner, editor. CERN Accelerator School { Fifth General AcceleratorPhysics Course. CERN, 1994.9

Thick Scatterers seen through the Twiss Functions

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