EFFECTS OF FRINGING FIELDS ON SINGLE PARTICLE DYNAMICS. M. Bassetti and C. Biscari INFN-LNF, CP 13, Frascati (RM), Italy

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1 Fascati Physics Seies Vol. X (998), pp th Advanced ICFA Beam Dynamics Wokshop, Fascati, Oct. -5, 997 EFFECTS OF FRININ FIELDS ON SINLE PARTICLE DYNAMICS M. Bassetti and C. Biscai INFN-LNF, CP 3, 44 Fascati (RM), Italy ABSTRACT eneal fomulation of magnetic field expessions valid fo multipoles of any ode ae epoted. Analytical expessions deived fom the cylindical model ae shown. The application on DAΦNE optics is descibed.. Multipola Magnetic Field eneal Expessions The scala magnetic potential geneated by a quadupola field not depending on z is: Pxy (, )= xy fom which by applying the gadient opeation the magnetic field components ae deived: B B x = y = y x Fo a quadupola field depending on z the condition of satisfying Maxwell equations in thee dimensions equies the dependence of the function not only on z, but also on : P(, θ, z) = xy = ( z, ) cosθ sin θ = ( z, ) sin θ!

2 whee z (,) must be of the fom: z (, ) = ( z) + ( z ) +... This fomalism can be extended to any multipole of ode m: m sin mθ Pm(, θ,) z = m() z + m() z +... m! [ ] Substituting sin mθ with cos mθ the multipole is skew. In paticula with cos mθ, m = the solenoid is epesented: P(, z) = ( z) + ( z) +... S S S The function m () z contains all the infomation on the magnetic field of the multipole, since the () z ae fom it deived: mp mp mp+ p () z = ( ) 4 dmp() z () z = dz p p m! d m() z p ( m+ p)! p! dz and theefoe the magnetic field of any multipole of ode m witten in cylindical coodinates is: sin mθ B(, θ,) z = ( m+ p) mp( z) m! cosmθ Bθ(, θ,) z = ( m )! sin mθ B (, θ,) z = m! p= p= mp z mp+ p= () z () z p+ m p+ m p+ m Fo example the cylindical components of a hoizontal dipole (m = ) magnetic field ae: { } { } B = z + sin θ ( ) ( z) B = cos θ ( z) + ( z) +... θ

3 fom whee the Catesian components ae: B = B cosθ B sin θ = ( x + y ) +... x B = B sinθ + B cos θ = + ( x + 3y ) +... y θ θ and the chaacteistic function is the vetical magnetic field on the axis: () z = B (,,) z y Fo a quadupole (m = ) it is the gadient on the axis: By ()= z x = while fo a solenoid the longitudinal magnetic field is the fist deivative with espect to z of the magnetic scala potential () z = B () z = S z ds dz The chaacteistic function of a eal magnet is in pinciple measuable.. Analytical Model (Cylindical Model) An analytical expession fitting the chaacteistic function of a magnet is a poweful tool fo any kind of analysis of the magnet chaacteistics and of its effects on the optics. This is what is obtained with the so called cylindical model : the field nea the axis of a multipole ceated by a cylindical sheet of cuents has an analytical expession defined by thee paametes: the length of the cylinde, Z L, its adius R and the ciculating peak cuent I c. The analytical desciption of m () z and its deivatives is possible. In the following table the chaacteistic functionss () z fo the solenoid, z () fo the dipole and () z fo the quadupole ae shown, using the definition of the functions: fh()= t t R + t h

4 Table I - Chaacteistic functions of some multipoles Multipole m m Solenoid Bz() z = S() z = µ oic Z f t z+ ZL () z ZL 4 L Dipole µ oic z+ ZL ( z) = [ f() t f3() t ] z ZL 4R Quadupole µ oic ( z) = 9f t f t f t () 8 3() + 3 5() 8R z+ ZL [ ] z ZL A quality of the functions f k + is that thei fist and second deivatives can be expessed though the same functions by the elationships: and g k+ dfk+ () t () t = = dt ( ) k + R R + t ( ) f 3 k () t d f dt k+ = ( 4k + k) f ( k + k + 3) f + k k+ + ( k + 8k + 6) f ( 4k + 8k + 3) f k+ 3 k+ 5 so all the highe ode tems ae linea combinations of f k + and g k +. Once measued the magnet chaacteistic function, it is possible to fit it with one o a supeposition of CM, using as fitting paametes the thee CM ones: length, adius and cuent. Fom thee on all field tems ae analytically defined. The example of the KLOE detecto solenoid is shown in the following figues and table. Thee CM with diffeent fitting paametes (see Table II) descibe the field on axis with good appoximation. In Fig. the dots epesent the measuement and the solid line the analytical fit. The goodness of the fit is shown in Fig. whee the diffeence between the longitudinal component of the magnetic field along z at two diffeent values of ( =., and = 7.5 cm) is shown: this diffeence follows the second deivative of the field on axis: B (, z) = ( z) + ( z) +... z S S3

5 Table II: CM fitting paametes fo KLOE detecto solenoid Z L (m) R(m) I(A) B (T) z Β (Τ) z (m) z (m) Figue : Longitudinal magnetic field Figue : Bz(,) z Bz(,), z =. 75 m in KLOE detecto solenoid in KLOE detecto solenoid 3. Effects on Linea Optics Linea magnetic elements at the lowest appoximation ae usually descibed by the ectangula model. In eality the chaacteistic function m () z is fa fom a step-wise function (see Fig. 3). A moe ealistic desciption is obtained by slicing the chaacteistic function, eithe measued o analytically descibed by the CM, into a easonable numbe of intevals 3, each of them descibed by a ectangula model matix (see Fig. 4). The poduct of all these matices epesents the total matix. Figue 3: Rectangula model and measued m Figue 4: Sliced computation of the measued m

6 In the case of a quadupole, the final matix can be witten again as a quadupole like matix, in which the two chaacteistic paametes (length and gadient) ae modified diffeently fo the two planes; the quadupole needs theefoe fou paametes. The diffeences between the ectangula model and the eal one ae of couse elated to the behaviou, which is stongly coelated to the atio R/ Z L. These diffeences ae consequence of the linea finge field and ae totally diffeent fom the non linea finging field effects coming fom the uppe tems ( (), mn z n > ). The linea finging field lowes the gadient in the focused plane and make it stonge in the defocused one. It is staightfowad, by applying: Q= KL 4π β to show that the total effect is a negative tune shift. In fact in the plane whee the quadupole focuses the value of K deceases and the coesponding Q is negative; in the plane whee the quadupole defocuses the value of K inceases and again Q is negative. The effect is of couse stonge in the plane with highe chomaticity. Fo example in the DAΦNE main ings acs, which contain two diffeent kinds of quadupoles, the value of Q is about. (H),. (V). Fo a solenoid, whose matix can be witten as the poduct of a otation matix and a focusing one 4 the method can be equally applied to the focusing matix, while the otation matix emains unchanged. 4. DAΦNE Inteaction Regions In the DAΦNE Inteaction Regions the magnetic field is a supeposition of quadupole and solenoid fields (see Fig. 5). K (m - ) B 6 z (T) z (m) Figue 5: KLOE half IR. Behaviou of low beta quadupole gadients and longitudinal magnetic fields of detecto and compensato.

7 The solenoidal compensation is done applying the Rotating Fame Method 4, in which the quadupoles ae tilted following the otation of the tansvese plane intoduced by the solenoids. The field components of a quadupole tilted by an angle θ T ae: 3 B (, θ, z) = sin ( θ + θ ) ( z) + ( z) +... T { } { } 3 Bθ(, θ, z) = cos ( θ + θt) ( z) + ( z) +... sin ( θ + θt) 4 Bz(, θ,) z = () z + 3() z +... { } Beams tavel off-axis in the IRs since they coss with a tunable hoizontal angle θ coss. Paticle tajectoies ae computed by integating the equations of motion along the IR. The linea jacobian aound the tajectoy on axis is in ageement with the linea IR tanspot matix computed with the sliced computation. Infomation about phase advance and optical functions at the IR ends ae deduced fom the jacobian fo the diffeent cossing angles. Fo the nominal optical paametes at the IP (β x = 4.5 m, β y = 4.5 cm), as θ coss inceases the phase advance along the IR inceases, especially in the vetical plane, this because aound the off axis tajectoy quadupoles add an altenate bending action, like a wiggle, giving vetical focusing (see fig.6). In the pesence of solenoids this focusing acts in the plane pependicula to the tajectoy plane point by point. At the IR end, whee the nomal modes become hoizontal and vetical because of the RFM method, the incease in phase advance appeas in the vetical plane..4.3 Y (mad) θ coss X Figue 6: Tune shift with cossing angle in DAΦNE optics

8 The R.F.M. applied on axis is exactly valid off-axis only in the linea appoximation. The fou paametes used fo decoupling the motion between the IP and the IR end 4, i.e. the thee quadupole tilting angles and the integal of the longitudinal magnetic field on the compensato solenoid, depend in pinciple on the cossing angle. Using the linea tems of the jacobian computed aound the tajectoy, it is possible to eadjust the values of the decoupling paametes by minimizing the non diagonal elements of the jacobian. The diffeences between the decoupling paametes with θ coss in the nominal ange have esulted smalle than the alignment toleances. Refeences. M. Bassetti, C. Biscai, Analytical Fomulae fo Magnetic Multipoles, Paticle Acceleatos, 996, Vol.5, pp.-5.. The KLOE Collaboation, The KLOE Detecto, Technical Poposal, LNF-93/. 3. C. Biscai, Quadupole modelling, DAΦNE Technical Note, L-3 (996). 4. M. Bassetti, M.E. Biagini, C. Biscai, Solenoidal Field Compensation, these poceedings.