1997 P a r t i c l e A c c e l e r a t o r C o n f e r e n c e, Vancouver, B.C., Canada, May 1-16, 1997 BNL-639 9 PARAMETERIZATION AND MEASUREMENTS W. Fischer and M. Okamura Brookhaven National Laboratory, Upton, NY 11973, USA Abstract Magnetic fields with helical symmetry can be parameterized using multipole coefficients (gn,in ). We present a parameterization that gives the familiar multipole coefficients (an,b n ) for straight magnets when the helical wavelength tends to infinity. To measure helical fields all methods used for straight magnets can be employed. We show how to convert the results of those measurements to obtain the desired helical multipole coefficients (&, &,). 1 INTRODUCTION The magnetic field inside straightmagnets can be parameterized in terms of multipole coefficients (an, b n ). We will present such a parameterization first. Fields of helical magnets (see Fig. 1) can be described in a similar way, by means of multipole coefficients (Tin, &). We give a notation for the (&, in) for which the (an,bn) are the limiting case when the helical wave length tends to infinity. A Cylindrical coordinate system (r,0, s) is used where s denotes the longitudinaldirection. We then assume that a magnetic field measurement device always parameterizes its measurements in terms of (an,bn), and give formulae to obtain the coefficients (&, g n ) when a helical magnetic field is measured. Three types of measurement devices are treated: rotating rdial coils, rotating tangential coils and rotating Hall probes (see Fig. (a), (b) and (c) respectively)..i MAGNETIC FIELD PARAMETERIZATION Straight Magnetic Fields In a current free region in vacuum where the electrical field I? is constant, the magnetic field can be derived from a scalar potential $ as B = -V$. We consider a magnet of infinite length, thus neglecting fringe fields. The symmetry condition of such an element is $[r,8, s) = $ ( r, 0, s+as) with As arbitrary. Therefore, the potential $ is independent of s, $(r,6, s) = $(r,6). Having a main fieldbo sin 0 the solution of the Laplace equation A$ = 0 can be written as B Figure 1: The conductor of a helical dipole (one helical wavelength long). This magnet will be used for proton spin manipulation in RHIC. (a)r d d coil (I,) r;mgrutial coil r (c) Hall p m h I Figure : Three methods to measure magnetic multipole coefficients. quadrupoleetc. ro is a reference radius. From B = -V$ and Eq. (1) the magnetic field can be obtained. We have B, =O. (4). Helical Magnetic Fields We consider again a magnet of infinite length and neglect fringe fields. The symmetry condition for a helical magnet The bn are called normal a d the an skew multipole coefficients. Here,the subscript 0 denotes a dipole, 1 a Work PerfOrmedunderW auspkof the US Depattmentof Energy Note that the European notation (see for example Ref. 111) The transformation is bn,ameriwn = bn+l,european a d an,amerrcan = differs &om the American one preseoted here. -Qn+l,Europeon. M B U T t O N OF THIS DOCUMENT IS UNUM-
is $(r,8, s) = $(r, O-kAs, $+As), where As isarbitrary. In other words, 0 - ks = const. k = 7rr/X is the wave number and X the wave length of the helix. k shall have a positive sign for right-handed and a negative sign for lefthanded helices. Introducing the new variable 6 = e ks, the symmetry condition leads to a potential @ which is only dependent on r and e, + ( r, 6, z ) = @(r,6). The tilde shall remind the reader of the fact that e -in a helix is similar to 8 in a ordinary dipole. Using ( r,8) as coordinates and having a transverse helical main Field BOsin B a solution of the Laplace equation A@= 0 is - [6n Os ( + 1 ) + sin ((. + ) )I (5) where In are modified Bessel functions and the fn are defined as = fn Z + + - l)! 1 rz kn * + I) + The in are called normal and the Z,, skew helical multipole coefficients (with respect to the main field Bo). The subscript 0 denotes a helical dipole, the subscript I a helical quadrupoleetc. TO is again a reference radius. The factors in ( 5 ) are chosen in such a way as to obtain the potential (1) when the helical wave length tends to infinity. ln this case k + 0, e + 0 and,the Bessel function can be approximated by in () N f (cf. Ref. 11). Now, the magnetic field can be computed as 5 + 1) ) + in sin ( + 1)8)], 1 Be = --Bs, kr (8) m ~s = -BO (7) fnln+l n=o ( + 1)kr) x + 1 ) I ) - 6, sin ( +,)I)]. (9) where 1; denotes the derivative with respect to the argument of the Bessel function. Since the Bessel function is nonlinear, a magnetic field with helical symmetry is nonlinear too, even the field of a perfect helical dipole. In addition, there is a longitudinal field component off the s-axis. Fig. 3 shows an example for the field inside a helical dipole. 3 MEASUREMENTS OF HELICAL FIELDS While the multipole measurements with rotating coils are obtained from the field within the coil area (shaded in Fig. ), the multipole coefficients of Hall probe measurements are obtained from the field on the circumference of a circle with given radius r. For rotating coils the magnetic flux is computed for both parameterizations (un,b,) -3 - Longitudinal position a [cml Figure 3: Magnetic field components along the s-axis with z% cm and y=3 1 cm for the k g n e t shown in Fig. 1. Note that end effects are not treated in this article. and (Gn, in). The results are stated in a form that allows a direct comparison. For Hall probe measurements the magnetic fields in both parameterizations are also stated in a form that allows a comparison. For all cases conversion formulae from (an,bn) to (z,,,?;n)are given in the same form. 3.1 Rotating Coils The magnetic flux through a coil is @(e) = N I B(r, e).da (10) where N is the number of coils windings. For rotating coils one has 6 = wt and the induced voltage U = -d@/dt is proportionalto the angular velocity w. 3.1.1 Radial Coils The area of a flat rotating radial coil ranges from rl to r and from s1 to s (cf. Fig. ). The magnetic flux (10) through the coils is Using (3) the magnetic flux (11) for straight magnets becomes With (8) the flux for helical fields is cm i p ( e ) = ~ -~. ~ d ~~~~~ n ~, ~ ~ ( ( n + l ) e ) - a n s i n ( ( n + i ) e ) ] n=o (13)
and in a helical field (cf. Eq.(8)) In (1 3) new magnetic multipole coefficients are used for which Sn(s1, s) = - and Tn(s1, s) = As + l)k sin + 1)kAs sin + 1)kAs (15) + As+ 1)k sin + 1)kAs cos + 1)kAs (16) with As = s - s1 have been defined. Jr 1; 1 R, = g f n l n + l ( + 1)kr). (1) The (6n, i n ) are again defined by equations (14) and the (S,,T,)needed in this definition by equations (15,16). 3.. Radial Field Components The radial field in a straight magnet is cf. Eq. ()) B,(B)= Bo n = O I(, [ancw(+i)o) +bnsin(+l)@)] 3.1. Tangential Coils The magnetic flux through this type of coil is (cf. Fig. ) *(e) = N with Br(r,0) rde ds. with Kn = (17) The difference A6 = O - 81 is fixed and one can assume that 61 = 8 - A8/ and e = 6 A0/ hold. With () the magnetic flux in straight magnets becomes (:)" 1 and in a helical field cf. Eq. (7)) + with Rn = fnia+l( 00 x + 1)kr). (3) Kn[ancos(+1)8) +bnsh(+l)e)] Also in this case the (hn,&.,) are defined by equations (14) and the (Sn,Tn)by equations (1516). rn+l 3.3 Conversion nr0 with Kn = ro" It is now assumed that a device parameterizes the measured magnetic field in terms of multipole coefficients (an,bn) for straight magnets. If the measured magnetic field has helical symmetry, the coefficients (&,in) can be derived by comparing(1) with (13), (18) with (19), (0) with (1) or () with (3). One obtains for all cases and for helical fields one obtains with (7) with R, = fn r Z A + ~( + 1)kr). The (in,in) are defined by (14) and the in this definition in (15,16). (19) With (14) it follows that (Sn,T') needed 3. Hall Probes For Hall probe measurements the magnetic fields, either tangential or radial, can be compared directly. Tangential Field Components The tangential field in a straight magnet is (cf. Eq.(3)) 3..1 &(e) = BO 00 n=o Kn[b,"l.(+l)@) -ansin(+l)@)] with K n = ( k ) n I 4 ACKNOWLEDGMENTS We are thankful to the members of the RHIC spin collaboration group for discussions. 5 REFERENCES [l] J. Rossbach and P. Schmiiser, "Basic course on accelerator optics", Filth G e n d Acoelerator Course, Univaity of Jyvi%kyE,Fiand, CERN 94-01 (1994). [] M.Abramowitz and I. Stegun, "Handbook of Mathematical Functions",Dover, New York (197).
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