Optimization of radial matching section 1

Transcription

1 Optimization of radial matching section Aleander Ovsannikov Dmitri Ovsannikov Saint-Petersburg State Universit (SPbSU) St. Petersburg Russia Sheng-Luen Chung National Taiwan Universit of Science and Technolog (NTUST) Taipei Taiwan Abstract Optimization approach to define geometr (parameters) of radial matching section in RFQ accelerator is suggested. This approach gives wider opportunities to design of radial matching section because it does not have certain prescribed laws of variation of focusing strength along the section. INTRODUCTION Acceptance at the entrance of RFQ accelerator as it is known depends on time and rotates with frequenc of RF field. On the other hand the input beam has constant emittance not changing in time. In this connection there is a problem of the beam matching with an accelerating RFQ channel. In work [] it was suggested to provide transverse matching of beam with the accelerating channel with the help of radial matching sections and concrete laws of change of focusing strength in these sections have been considered. The net ears this problem was considered also b different authors [-4]. In the paper the new approach for the solution of the given problem is suggested. Parameters of radial matching section are defined as a result of the solution of some optimization problem. The work is done under support of the National Taiwan Universit of Science and Technolog and Saint-Petersburg State Universit Joint Research Project No. RP7-5

2 PROBLEM STATEMENT For radial matching section of accelerator charged particles dnamics in ( ) plane which is perpendicular to longitudinal ais in the case of microcanonical charge distribution can be described b the following equations [5]: d eu = W a ei r r cos W πε vr r r r + ( θτ + ) = Q ( τ r r ) () d eu = W a ei r r cos Wπε vr r r r + ( θτ + ) + = Q ( τ r r ) () where τ = сt θ = πω c U is intervane voltage W is charged particle rest energ ω is accelerating field frequenc is initial phase c is light velocit a is radius of channel v = z& is a longitudinal velocit of particle which is constant along matching section r and r are normalized beam envelopes I is beam current. The equations () () can be transformed in sstem of the equations d ξ dη = A ξ = A η (3) where ( ) ξ ξ look like ξ = ξ = d ξ = η = = d η η = and matrices A and ( η η ) A A = Q ; A = Q. (4) Let the set of conditions of sstem (3) fills in planes d d and during some moment τ ellipses correspondingl and det det =. Then matrices and = ξ ξ η η (5) satisf to the following sstem of the matri equations

3 d = A = A A A d. (6) The sstem of the equations (6) is solving on interval from the input in regular part of the accelerator to the entrance in radial matching section i.e. from τ = Τ toτ =. Initial conditions for sstem (6) are the matrices of ellipses defining acceptances of a regular part of the accelerator depending on an initial phase : ( Τ ) = ( ) ( Τ ) = ( ) Τ. (7) Optimization problem for radial matching section is to find function a ( τ ) Τ i.e. law of the radius change along the matching sections providing under the conditions (7) the maimum possible overlapping of families of ellipses at the entrance of radial matching section. where Let's consider functions B Φ ( ) = Sp( ( ) B ) и Φ ( ) = Sp ( ) ( B ) (8) B are given matrices Sp is the trace of corresponding matri. Functions ( ) Φ and ( ) Φ characterize deviations of ellipses and at τ = accordingl from the given ellipses B B. Introduce the functional J ( a) = c Φ ( ) d + c Φ ( ) d (9) estimating degree of mutual overlapping of ellipses corresponding to various initial phases at the entrance of matching section. Here and are limits of variation of initial phase ; c c are some positive constants. METHOD OF SOLUTION Let's present elements of matrices ( ) and ( ) τ τ as follows 3

4 + = α α α + = α α α () where = r = r. Let's notice that values α and α are so-called Courant- Snder or Twiss parameters [6]. Now the equations (6) can be written in a following form d = α d = α dα = Q dα = Q ( τ r r ) ( τ r r ) + α + α () Rewrite the sstem () as a sstem of the equations of the form σ = f ( τ σ u) () d with corresponding conditions on the right end Here ( σ σ σ σ ) ( Τ ) = σ ( Τ ) [ ] σ σ = 3 4 σ = α σ = 3 (3) σ = α σ = 4 ; a( τ ) u = is a control function; initial phase value = belongs to some interval. The functional (9) we can write in the form We introduce vector-function ( τ ) differential equations with initial conditions ( u) = g( σ ( ) ) J d. (4) ψ satisfing on the trajectories of sstem () the sstem of ( τ ) f ( τ σ ( τ ) u) d ψ = σ ( σ ( ) ) g ψ ( ) = σ Then the variation of functional (4) can be represented [78] as δj T ψ (5) [ ]. (6) ( u u) = ( p( τ )) u( τ ). (7) 4

5 Here vector-function p ( τ ) is epressed as follows where vector-function ( τ ) (5) with special initial conditions (6). ( τ σ ( τ ) u) f p ( τ ) = ψ ( τ ) d. (8) u ψ is the solution of the auiliar sstem of the differential equations Vector-function p ( τ ) can be used as a «direction» (anti gradient) for minimization of functional () in space of admissible controls u ( τ ) (for eample continuous functions that have their values in some bounded compact set). We set where > control [8] ε ( τ ) u( τ ) p( τ )ε u = + (9) ε and u ( τ ) is variation of control ( τ ) ε u. In the case of smooth control functions it is possible to consider following variation of u ( τ ) = u( τ εδ ( τ )) [ T ] ε + τ. () Here ε [ ] is a parameter; δ ( τ ) is a smooth function such that τ + εδ ( τ ) T. Then obviousl u ( τ ) is alwas an admissible control we have u ( τ ) = du( τ ) εδ ( τ ) + o( ε ) ε variation of functional (4) can be written down in a following form and the δj = ε T ( p( τ )) ( τ ) du δ ( τ ). () This representation of functional variation also can be used for minimization of functional () and gives obvious hint at function δ ( τ ) selection. Obtained analtic representations (5) () of variation of functional (9) were used to find geometric parameters of radial matching section of RFQ accelerator of protons (initial energ 45keV output energ 5 MeV intervane voltage kv RF field frequenc 35 MHz initial cell length 3.9mm). One of possible choices of law of variation of channel radius along radial matching section is presented on the figure. At the figure RFQ acceptances without radial 5

6 matching section are given. And illustration of radial matching section function is shown at figure 3. 6 RADIUS OF CHANNEL IN RADIAL MATCHIN SECTION 4 R mm Z mm Figure. ACCEPTANCES WITHOUT RADIAL MATCHIN SECTION (XdX/) ACCEPTANCES WITHOUT RADIAL MATCHIN SECTION (YdY/) dx/ mrad -.5 dy/ mrad X mm Y mm Figure. ACCEPTANCES WITH RADIAL MATCHIN SECTION (XdX/).5 ACCEPTANCES WITH RADIAL MATCHIN SECTION (YdY/) dx/ mrad. -. dy/ mrad X mm Y mm 6

7 Figure 3. CONCLUSION New mathematical models and methods of RFQ structure optimization were suggested in works [9-]. In this paper optimization approach to find geometric parameters of radial matching section is suggested. The mathematical model of optimization with use Courant- Snder parameters is developed and analtical representation of gradients of functional to be minimized are found. It allows developing different methods of directed optimization. Also it can be used for calculations of tolerances of geometric parameters of the channel. REFERENCES. Crandall K.R. Stokes R.H. Wangler T.P. RF Quadrupole Beam Dnamics Design Studies. In: Proceedings of 979 Linear Accelerator Conference Montauk 979 p. 5.. Tokuda N. Yamada S. New Formulation of the RFQ Radial Matching Section. In: Proceedings of the 98 Linear Accelerator Conference Santa Fe 98 p Balabin A.I. Kabanov V.S. Kapchinsk I.M. Kushin V.V. Lipkin I.M. On beam matching with RFQ channel. Journal of Technical Phsics 985 Vol. 55(3) p K.R.Crandall RFQ Radial Matching Section and Fringe Fields Proc. 984 Linac Conference SI Darmstadt Report SI-84- p.88(984) 5. Kapchinsk I.M.. Theor of linear resonance accelerator. Moscow Enrgoizdat 98 p M. Reiser Theor and Design of Charged Particle Beams. New York 994 p Ovsannikov D.A. Modeling and Optimization of Charged Particles Beam Dnamics. Lenigrad 99 p Ovsannikov A.D. Control of program and disturbed motions. St.Petersburg Vestnik SPbU Vol. (4) 6 p.-4. 7

8 9. A.D.Ovsannikov B. I. Bondarev A. P. Durkin New Mathematical Optimization Models for RFQ Structures Proceedings of the 8 th Particle Accelerator Conference New York USA 999 pp Ovsannikov D.A. Ovsannikov A.D. Svistunov Yu.A. Durkin A.P. Vorogushin M.F. Beam dnamics optimization: models methods and applications // Nuclear Instruments and Methods in Phsics Research section A 558 (6) pp.-9. 8