Lawrence Berkeley Laboratory UNIVERSITY OF CALIFORNIA

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1 e r LBL-3875 UC-44 Lawrence Berkeley Laboratory UNIVERSITY OF CALIFORNIA Accelerator & Fusion Research Division The Vector Potential and Stored Energy of Thin Cosine(n) Helical Wiggler Magnet S. Caspi December 995 I 5 Prepared for the U.S. Department of Energy under Contract Number DE-ACO3-76SFOOO98

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3 SC-MAG429 LBL-3875 The Vector Potential and Stored Energy of Thin Cosine(n) Helical Wiggler Magnet.*. Shlomo Caspi Lawrence Berkeley Laboratory University Of California Berkeley, CA 9472 December 7,995 * This was supported by the Director, Office of Eneigy Research, Office of High Energy and Nuclear Physics, High Energy Physics Division, U. S. Department of Energy, under Contract No. DE-AC3-76SF98.

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5 Abstract Expressions for pure multipole field components that are present in helical devices have been derived from a current distribution on the surface of an infinitely thin cylinde8 of radius 2. The strength of such magnetic fields varies urely as a Fourier sinusoidal series of the longitudinal coordinate z in proportion to cos(n8--wmz), where w, = 'f2m-r L, L denotes the half-period and m=,2,3 etc. As an alternative to describing such field components as given by the negative gradient of a scalar potential function (Appendix A), one of course can derive these same fields as the curle of a vector potential function A' - specifically one for which V x V x A' = and V A' =. It is noted that we seek a divergence-free vector that exhibits continuity in any of its components across the interface r=r, a feature that is free of possible concern when applying Stokes' theorem in connection with this form of vector potential. Alternative simpler forms of vector potential, that individually are divergence-free in their respective regions ( r<r and r>r), do not exhibit full continuity on r=r and whose curl evaluations provide in these respective regions the correct components of magnetic field are not considered here. Such alternative forms must differ merely by the gradient of scalar functions that with the divergence-free property are required to be "harmonic" (V2q= ). A summary of the vector-potential derived in part one is given below. In part two we derive the magnetic field components and check the validity of V A' =. In part three the stored energy is derived from the vectorpotential (shown below) and finally in part four we reduce the problem dimensionality to the more familiar 2D results by extending the period to infinity. and the stored energy density Where In and Kn are the "modified" Bessel function of the fist and second kind of order n, aqd the prime denotes differentiation of the Bessel function with respect to its argument. An alternative form for expressing the vector-potential as Bessel functions and their derivatives of order o&, is given in the text. Magnetic Field Components in a Sinusoidally Varying Helical Wiggler, LBL-35928, SC-MAG-464, July 994.

6 Analysis One may consider a direct evaluation of the vector-potential function through use of the integral formulac with subscripts on the coordinates to indicate source-point locations. The integration in 3 is taken to extend from --co to + co and the integration in 8 to extend over an interval of 27r. We have undertaken such an.evaluation, using for the source an expression consistent with that cited previously in Ref? : Wm = ( l).r (> Gn,m = n!rn and L n wmr B = B, dipole field 2B2 = Gm quad gradient 3B3 = Sm Sextupole... Bn,m The pair of current density components satisfy the conservation condition V - 2 = We may put $ % + $% = as required. = 62 ioo = - sin ~ cos 6% + = - sin Oo(cos e$ - sin 68) cos &(sin 6,. = sin (8-6)$ cos (8-6o)ie + + cos 66) and introduce working variables t = 6 - and s = zo - z for the purpose of performing the integration. The z component of ' A' The z component of the vector-potential may be written as : 4.r -- cos (neo- wmzo) R d B o dzo ngnlmw m R 2 K ( w m R ) JB2 +,.2 + ( z - zo)2 Employing the new working variables and the relation, Panofski and Phillips, Ed.2, Eq. (7-42), p Rr cos (e - o)

7 we may alternatively write, and proceed with the evaluation of the two double integrals. First double integral For the first double integral, by referenced and with the understanding that odd functions integrate to over foo 77 -w cos (nt - w,s)dsdt dr2+r2+s2-2rrcost = 77 -OJ cos nt cos (wms)dsdt dr2+r2+s2-2rrcost 2* w./ -w sin nt sin (wms)dsdt dr2+r2+s2-2rrcost + r2-2rr cost) cos (nt)dt + In recognition of the summation theorem * and of the orthogonality propemes of circular functions and since I-, = In and K-n = IC, the above expression for the double intern. can be reduced to (for rlr Second double integral 77 Similarly, we show that the second double integral vanishes. -w sin (nt - wms)dsdt dr2+~2+~2--2rrcost -OJ sin nt COS (wms)dsdt dr2 r2 s2-2rr cost + + -w e I.S. Gradshtyne and I.M.Ryzhik, Table of Integrals..., Eq (2). p.49 G.N. Watson, Bessel Functions, Sec..3, Eq. (8) p.36, with n=o.. Abramowitz and S t e p, Chapter 9, Eqs , p cos nt sin (w,s)dsdt dr2+r2+ s 2-2 R r c o s t

8 In recognition of the summation theorem, the orthogonality properties of circular functions and I-, = In and I{-n = ICn, we may write for rlr 72 + r2-2rr cost k=-oo =4 f I{k(WmR)Ik(wmr)COS ( k t )sin (nt)dt Kn(wrnR)In(wmr)cos (nt)sin (nt)dt= The expression for the vector-potential may now be written for rlr as, and for and s,we may interchange the arguments of the Bessel functions and write, wiq evident continuity at the interface r=r. The r component of A In developing the radial component of the vector-potential we shall employ similar technics to those previously applied for evaluating Az. The r component of the vector-potential may be written as : with cos (ne - wmzo) = cos (ne - wmz) cos (nt - wms) - sin (ne - wmz) sin (nt - wms) and using the working variables as before, we may alternatively write : 4 9 = -4n n=l m=l I{;:ZR) [ - cos (ne - wmz) J J COS (nt - wms) sintdsdt JR2 + r2 + s2-2rr cos t sin (nt - wms) sin tdsdt + sin (ne - wmz) / / d R 2 + ~ 2 + s 2-2 R r c o s t First double imgral We show that the first double integral in the above expression, vanishes. With the understanding that odd 4

9 functions integrate to over 3x, 77-3 (nt - wms) sin tdsdt dr2 r2 s2-2rr cos t COS + + = IJ 2* 3-3 COS nt COS (Urns)sin tdsdt + dr2+r2+s2-2rrcost 77 -m sin nt sin (Urns)sin tdsdt dr2+r2+s2-2rrcost + r2-2rr cos t) cos (nt)sin tdt + and in recognition of the summation theorem, the orthogonality properties of circular functions, the identities I-, = I, and IC-, = ICn and the fact that + cos nt sin t = -[sin ( n )t - sin (n - l)t] 2 the above expression for the double integral is identically, since drz+ r2-2rr costcos (nt)sin tdt = 2 =4 Second double integral Similarly for the second double integral : -m sin (nt - wms) sintdsdt R2 r3 s2-2rrcost ICk(wmR)Ik(wmr) COS ( k t )COS (nt)sintdt k=--3 ICn(wmR)In(Wmr)COS (nt)cos (nt)sintdt = sin nt cos (wms)sin tdsdt dr2+r2$s2-2rrcost cos nt sin (urns)sin tdsdt dr2+r2+s2-2rrcost dr2+ r2-2rr cost) sin (nt)sin tdt + In recognition of the summation theorem, the orthogonality properties of circular functions, I-, = I, and IC-, = IC, and the fact that sin nt sin t = --[cos (n )t - cos (n- l)t] we may write for rs, r2-2rrcostsin(nt) sintdt = 2 72 ICk(wmR)Ik(wmr) cos k=-3 (care has been taken to verify that the above is true for the case n=l as well) 5 (it)sin (nt)sintdt

10 The final expression for and for the region Xr in the region can now be written as s, we interchange the arguments of the Bessel functions and Write, with evident continuity at the interface r=r. Alternatively with the relation : we may express the vector-potential in terms of Bessel functions and their derivatives of order n only. The component of A' The vector-potential in the direction is : Wlth cos (ne - wmzo) = cos (ne - u m z ) cos (nt -urns) - sin (ne - wmz) sin (nt - urns) and use of the working variables, we alternatively write : ' --E (nt - wms) COS tdsdt dr2+ r2'-+s2-2rr cost sin (nt - urns) costdsdt + + ~2-2Rr cost COS = 4T n=lm=l. First double integral For the h t double integral and in recognition of the "summation theorem" and that odd functions integrate to over =too (nt - wms) COS tdsdt = R2 r2 s2-2rr cost COS -m + + 7/m -m cos nt cos (Urns)COS tdsdt dr2 r2 s2-2rr cost COS + + (nt)cos tko 6 ( m sin nt sin (wms) cos tdsdt dr2+~2+~2-2rrcost JR2 + r2-2rr COS t ) dt +

11 .. Again, using 'the orthogonality properties of circular functions, the identities' I-, = I, the fact that,i<-, = I<, and cosnt cost = -[cos ( n )t cos ( n- l)t] the above expression for the double integral reduces, for e,to Second double integral 77 Similarly, we demonstrate that the second double integral vanishes, -w sin (nt - wms) cos tdsdt dr2 r2 s2-2rr cost + + -co sin nt COS (Urns)COS tdsdt dr2 r2 s2-2rr cost w cos nt sin (wms)cos tdsdt dr2 r2 s2-2rrcost r2-2rr costsin (nt)cos tdt + and in recognition of the orthogonality properties of circular functions, I-, = I, and I L n = If, and the fact that + + sinnt cos t = -[sin ( n )t sin (n - l)t] 2 we may write for rsj, + r2-2rr cos t) sin (nt)costdt = 2 The final expression for & in the region 72 I<k(wmR)Ik(wmr) cos (Ict)sin (nt)costdt k=-w can now be written as 7

12 and for the region s,we may interchange the arguments of the Bessel functions and write i with evident continuity at the interface r=r. Alternatively with the relation : we may express the vector-potential in terms of Bessel functions and their derivatives of order n only. The Magnetic Field Components We shall proceed and derive the magnetic field directly from the vector-potential, a process that may as well serve as a check for such a field when compared with similar results obtained from the scalar potential as shown in referenceb.accordingly, The z component of B' With the vector-potential derived earlier, we proceed in deriving the field expression in the region 8 e

13 Applying the relations : ";++ 3A-- + ( n )In+= XIn ( n - )In-= "In I(n+l+ Ii;2-= - 2 ~ ; (where x corresponds to the Bessel function argument), the squiggly brackets reduces to : -2(wmr)I';In is : the field in the region n=l m=l Similarly applying the above procedure to the region e, as it showed be. The r component of B' We continue and derive the radial field expression in the region I S J, Therefore, 9 and

14 (with two different arguments) the radial field component in the region e is : I n=l m=l and similarly applying the above procedure to the outer region as it showed be. The 8 component of a we get, B' - In deriving the azimuthal field component in the region B, Therefore, x COS and introducing the relation (ne - wmz) I<n+l(wmR)In+l(wmr) - I{n-l(wmR)In-(wmT) the field component in the region = is : n=l m=l and similarly by applying the above procedure to the outer region as it showed be. we get, as it should be.

15 2 - a check As a final check, we shall show that the divergence of A' vanish throughout space including r=r, ' The divergence of lda8 daz =O ar de az -, d(ra,) V-A=-- We first check the divergence in the region Therefor : Applying the identities with different arguments a,b :.

16 The stored energy in multipole helical windings Now, it is only a hop skip and a jump to apply the vector-potential in calculating the stored energy in helical devices. From the definition of the energy, We recognize that we need integrate the vector product over the current surface only and divide-thestored energy by the volume of integration taken here as extending over the period 2L. (the current density is generally per unit area but when applied to thin windkgs is per unit length, the energy T-A density is, J = m.). The most general expression for LJ J E 2* e=~ L ~ L OP- Lo n 3-A on the surface r = is~ : cos (ne - wmz>cos (io - u p >x i i (w,r2)(wjr2)i<hi<j nign7mgi7j m We shall omit writing the argument wmr in both Bessel functions I and K. Making use of the orthogonality properties of circular functions, ii COS (n9 - wmz) COS (i9- wjz)rdodz = {ERL ; n = i, m=j -L the only terms that due not vanish, are for n=i and m=j, therefor : The term in the above bracket may be reduced by applying the relation : resulting in : e=--xx I 2PO n 2 m G,mC R2 I<: i i n f i m f j

17 We express the energy density in terms oe current density using the relations from referenceg a. J t = Joe cos (ne - wmz) where denotes the direction of current flow in the helix. In terms of the total current per pole Ipole, we have : and inversely we may write, so that the energy density can be written as : The limiting 2 dimensional case As a farther simplification and a check, we reduce the results obtained for helical devices by extending the periodicity to infinity, limlc,,wm =, and compare those with more familiar 2D cases of multipole magnets. with sj o g Forces in a Thin Cosine(n) Helical Wiggler, LBL-36988, SC-MAG-495, March

18 The 2D vector-potential reduces to : - for r 5 R A, = Bnrn cos ne and the stored energy is : Example - dipole, n=l We have calculated and plotted (using mathcad) the magnitude of the vector-potential components for a dipole and a half period length of LF2. cm. As a parameter we varied the n=l, with a single period m=l, w l = winding radius, R=l.O,.5, 2., and 2.5 cm. z, 4

19 .25.2 (G-crn) Ae (G-crn) A z.3 (G-cm) Figure The magnitude of the three vector-potential components in a dipole helix with a period 2M.O cm,zfo (for A, 8 3 and for A8 and Az at O=O) 5 5

20 Appendix A Field Components The field components, derived from a scalar potential, in the region interior to the windings referenceb>: dv = B, = -Gn,mwmIi(wmr) sin (ne - w m z ) dr n=l m=l The field components in the region exterior to the windings r>r are : 6