This paper describes the path of charged particular in inhomogeneous fields, having information of magnetic field at every point, and adapts the methods for a high transmission double focusing spectrometer intended to enable focusing and selection of positrons for the measurement of Bhabha's exchange parameter using position electron scattering

Bhattacharyya, P. K.

English

Defence Science Journal

GEOMETRICAL RAY TRACING OF ION-TRA~JECTORIES FOR AN ASTIGMATIC SECTOR POWBED BETA-RAY SPECTROGkA4PH 9 High Altitude Researoh Laboratory, Gulmarg, Rashmir (Reoeived 26 August 1970) Thia paper desoribes the path of charged partiales in inhomogeneoue fields, having information of magnetio field at every point, and adaptsthe method for a high transmission double foou~ing apeat- rometer intended to enable f uaing and selection of positrons for the measurement of Bhabha's @=ohan@ pmm- o h g  Pa8$ron-e1eBtron scattering. In beta-ray spectroscopy the demands for large solid angles are usually more pro- nounced, particularly in coincidence spectroscopy, using radio-active sources. To achieve a, good luminosity alongwith a good resolution one is restricted to accept more incident flux in the median plane, rather than that in the vertical plane. Usually it is d%cult t o  m&zh with the theoretical field profile for n better resolution, when one uaes larger polewidth, to accomplish greater transmimlon, alongwith a wide- zap. Iu this paper, it is. intended to give i geometrical ray tracing method, which can k; used safely, for comp%tible reqwements of polewidthand polegap, for sector fields double focusing type and orange spectrographs of the type given by Nielsen b Kofoed- 'if nsenl. The particular virtue of this tracing system is that it can analyse several such # ~ s  which accept a large angle in the vertical plane, by adding them together to increase ians'mission in the s a w  proportion a t  a given resolution, using any suitable rotationally ymmetrio fieldz's. Many of the imaging relations, eg, as derived by 8turrock4, till .k~ow have been dealing mostly with the properties of image forming by paraxial ray assemb- - lies; where the field is assumed to be governed hy the well-known relation BI= B, (r/R?-n , "  [where Bo is the vertical component of magnetic field a t  bending radius r, and B, the a magnetic field a t  the central ray or reference orbit of radius R, whereas n is the negativc exponent of radial dependence of magnetic field (-TIB) (d B/dr)] for point sources only near the oentral ray or reference orbit. The proposed geometrical ray tracing solution bas been found useful and free of such limitations for an accurate determination of imaging relations. It will be recalled from generalized Barber's rule of magneto-optics that the bending edge; source and the image must lie on the eeme straight line. In  other words, determi- nation of object and image distances to a sector field (Fig 1) can be attributed to analytic properties of circles and straight lines, and a geometrical image formation technique given by Siegbahns ctan fairly accurately describe the state of affairs except when the space- ? obrge is predominant. For an easy short-cut Lavatelli et al? and Fowler et a].' have been, as such, satisfied with two dimensional graphical analysis8 and in a typical example, I 284 Dm, Sor. d.. VOL. 21, OOTOBEB 1071 an ion-beiim in the shape of a narrow ribbon has been chosen, by Robinsons, to solve the complexity, so that its dimension parallel to the magnetic field (here called h )  can be assumed to be large in 'compari~on with the maximum thickness of the beam (Fig 2) while trs versing an 180" sector spectrometer. (a) P A R T I C L E  TRAJLCTORIES ( ~ 1  GLNCRALIZCD BARBLO'S RULE Using Newtonian mechauiw (emu), the force on the particle connects tlie image formation when expressed as, mv2 - = Bev f 2 (* lecalvh) (PIX) r Fig l-~adial focmiing in a sector fooused magnetid. wa being the mass in gm and v the velocity in speotrometer. cn~/sec of the particle moving in a cyclotron radius r, inside n magnetic field ZZ, perpend;cular to $s motion, within the im-Eihbon carrying a current I. The sign refers to particles which ere respectively closer and fafther from the mean curvature, assuming a as the half angle of divergence of the beam, and ,3 t,hp location of the particle in the laminurn by  measurement d ~ t s  angular position from the centre of the beam. We can find geometrically, for a constant bending spec- trometer (R-constant), that (Fig 2) in situations of acceptance angle 2 a < 2 sin-1 (1/6), (or for the ion-beams of sngular spread -0.09 sterwd), ions reeo1Ve t,hem@lves into a single beam (with only first order aberration having major contribution) for bending radinr of 30 cni. The centre of curvature C' and CI of the rays above and below the centrb orbit are found to lie very nea,rIy in the same vertical line passihg through O. This roa- assumption although sufficient, perhaps would break down for very large (radius of CL vature) high-transmission spectxometers, where it has been found that second order abt rations, which are function8 of radial and axial maximum aperture widths, contribu: much. In  such a special case, the method wo~ildnot point out any real improvement i resolution, because it uses only first order properties of focusing. Since general equations.. of motion are derived in many books on electron optics, the procedure outlined will merely ' be for nornhal entry and exit cases in order that theoreticalaspects of the problem nlay . be established only. Further, e%ects of source width and the resultant image shapes, are not presented here on grounds of simplicity. U N D E R L Y I N G  T R E O R ' Y  Every arbitrary spatial trajectory or positron ray emitted from the source is assumed to be circle inside ' the field space, crested by the seotionof the plane containing the trajectory, a t  an arbitrary angle, with the sphere whose centre lies on the line passing through the bending edge and perpendicular to , rf Barber's axis (ie, the line joining the source, i.. bending edge, and the image). We learn, therefore, that high energy particles would ~ i g  %-Two dimemima1 ray trsoing. \ L ~ T T A O H I B I ? A  I Chmetrioal Ray W i n g  of 1on.Trejectories 285 have to travel a circular trajectory within a sphere of greater radius and centre shifted upwards from the Barber's axis lSOl in - Fig 3, and the low energy , particles would move within spheres of smaller radius and haying aentre shift- ed downwards in the same vertical line. r For an astigmatic sector focused nlagnet c+,r.r,  rPklar all these spheres would cross-over each other on a mmnlon circle, which would contain the Bprber's axis' as ' diameter, in Fig 3-Three dimension anelytic diagram used the same plane, which could be named as for ray tracing. Barber's plane. ,411 these trajectories can be safely (for first order focusing) imagined as sign curve8 of various aaplitudes relative to the "stretched-out", central ray, or reference orbit fol~tical axis bf the system). To dis- cuss the app~mtion"of these postulates we Tvould first give the detaile of the coordimte system. A a~niple mechanical armngement is chosen ,(Fig 4), where, the end faces of the bending magnet are ~erpendicular to the+ optic axis and the beam passes symmet- rically through it. geleating the central ray as a ray that has a given position and direc- tion at  the input at a given momentum,-we define the entmnce coordinate system as (XI, Y,, Z1) with migm on the central ray', X-d~rection along the ray, Y-direction nornlal to it ~ n d  in the median (about which the complete magnet system is mirror sym- metrit), and Z-direction perpendicular to the median plane. Similarly, in the exit space, , we place a right &&cfd Q O Q E ~ ~ %  system (X2, Y2,Zz )  where again the X-direction coin- - cides with the central ray after its defleotion through the magnet system. Let us now determine the position and direction for an arbitrary ray in the exit space for which coor- h a t e s  are (XI, Y,, Zl), the momentum spread dz1 8=~p/p,anddopeey;=~1mdz;= a x  s., ,'/ \./pa '4, , ClELD SPACE In other words we seek the ha t i ons  at Y = 1 1 l '  1 3' 6) (a) 2s = f* (~19 yl', Z1,~I'Y 8) (4 I % d I @ THE * & P E R  Z-DlRhCTlON OUT OF THE PCANE OF %dth entrmoe ooordihate (XI, r, &,) and exit d n a t ) r  (X* Ye ZA. w'here t e  &myme that the input parameters yl, y: and 2, and z,' are speoified a t  the {oint where the ray psses th8 plane xl= 0. The parameter S =A p/l)' is a constamb for the motion in s magmt system with stationary field, On6 can assume withqut muah loss of generality (@.A. Znge-private cm~l~unication) that deviations from tho central ray are generally small so that y,', z( an& S are much smaller than unity and that yl & a, both are small con ip~ed  to IE; tBe2radius oi bending of the central ray. With above assumptiofis it is appropriah ta expnd  equation (1) in Taylor senes. We write the series in the following form far equation l(a). * higher-order terms are neglected. The partial derivative?, given in the  paxw~ftbsis, ape" equal- te zei& bomm olE the symmetry about the median plane which t&e magnet-i~ assumed to possess. Simp- lifying the above equation, one gets, The expansions have been carried only to second-order. The partial dkvatives that appear in Taylor expansion can be wribteninabbreviated form using En~ae*s notationB and can be called as "focusing coefficients". These paramths  Bre fti&& made non- dimensional by dividing yz and 2, with the reference radius R. - - ,  .. / Y1 y21R = (~19)  + (y1yf) Y,' + ( ~ 1 8 )  S + (9/Y2) ( ~ 1 l R ) ~  + b19y1y') g 9,' + (y/yS) % 8 + j HOT example, the meaiting of ssme coefficients in equation @) in'temfi, of AQ@&OI~ ' , - .  I . % . - I -  - i -- I > ;* .'*. - *  - "- I used $ zegular series i~ {y jy )  = ag2jayii (yly ')  = B R2 a2yz ayl' (Y/Y") ~ = g . * 80me coefficients eg (gb!'yz) = 0. S i m k l y  expressing other functions, L B' = (Y'lY) (YlIR) + (Y'IY') yl' + (Y'lS) 6 + tY'lY2) ( ~ 1 i R ) ~  + (Y'lYY') (tyllR) ~1'- + (y11y8) (y,lR) 6 + (Y'IY'~) + (Y'lY'S) 91'8 + ($'IS2) s2 + ( ~ ' 1 2 ~ )  (z11Rl2 + Cy'lzz') (z1lR) ( Z l ' )  + ( Y ' I ~ ' ~ )  z,'2 (3) 2-2/12 = (z i t )  (~1lR3 + (&'I 21' + ( 2 1 ~ ~ )  (YIIR) (ZllR) + ( 2 1 ~ ' ~ )  YI' ( 2 1 1 4  + (ZIYZ')  ( Y ~ I R )  2;. + (z&J y3'z1' + ( z l k ; )  82; '; (4) I '  " %,* 1 + 1 1 I (9;IR) I + / Y  y1' ( z , / ~ )  + (2.18z) 8 (z,lR) + (z'lyz') 2,' (y,lR) + (z'ly'z') y,'~,' + (z1/Sz') Szl' (5 )  * as ~ l e  g& -the exit parameters andn knowledge about foousing an a ma,gnet system whose field can be described by the ex* pressio ' (6) field coefficientis n, @ and Y are functions of geometry end permeability of the spectrometer's ple-pieces &&* 3 b r  e&nt to that sf the yoke. Wikh the exit para- meters defhe%,"*e would proceed to determine the ray hebviour in space using assump- tions postulated before alongwith generalized Barber's rule -of magnetq-optics. Let us choose an arbitrary ray forming a part of ..I circulltr hrajectory conkined in $he dotted sphere of radius r ,  within the magnet system. Equation of such a, sphere with its centre shifted upwards by a distance $ from the nenke of the sphere containing the central ray (010,' = E )  is, J (x+f sin Y ) 2  + (y+R-,$ cos y)2 + z2'=)2 (7) -. ?&&2er EQW the plane through the Barber's axis and let it be assumed as Barber's plane, given by the expwt)6ion, fy+B).eos y = x sin y, 2 = 0. lane making an angle 1,4 with X2 Y ,  pjane can be expmssed as - y + R - . c t a n y + l z = O ,  wh~~el=cos t ,h .  The intersection point PA" (0, y2, %)^ of the arbitrary circular orbit with plane x2 -- 0, -9 be put as 5. I i.C. T - .  , 5- ' I p WB X j- (- sin2 --62 + r 2 + + ~ z z ) +  .z2=-- , i - ' 1 4 z2 would give 1 .  - -, a r *  I * < - @ 9 - @ = ,/j ( 1  +la) : *tan y (*12+(h> : tao y (Z,/+JZ) & as ds ' as e ~ p P j j  6- +-Lf ' , Tangent at PWd (0, ya, 3 ie<givee by w A . ,  +s2 3-0  3- _=i-= P, (8) Now in order to evaluate the Pocushg properties of rhe magnet system we ;fid -&& intersechion of the -tangent aith the plane p , I I e l  to Y,ZA %Jane, given by z q$@i~t The point I, where the oentralrayaeets the hag0-ph&lane and the ~ ~ b e r y s  sldb has co dinstes (R 00% 7, 0, 0).  The coordinates of 5'' is given as, i BHATI~ACEABYA i Uee$etfoai ,Ray %o&g of ;Isne%ojectoriee PBQ , , x = R cot y b *Y = .!I2 4- C b n  Y (!712+4i ) From the equation (8) we get by proper substitutioa, C d x  (1 + 12) = R cot y or C = R cot Y]l(i (1 + $2) .',,d v , y=-B+ d R c o t y  *1+2 + d m  &*-a r C+J2+di) - , = 1 (!I ---45) (RE + dij 1 d h  (l-kt2) and T. z = zz + C tan (J<~---ql) A *;< $1 76 U-i">%* . - * ,  2-: *- c 1 ' - as bent and seleched by a. sector where Dp is the momentum dispersion accepted by the magnet system in order to converge the arbitrary ray. The resolving power (RP), can similarly, be given as , RP = p l n p  = R D,/IIW If the magnet has a low momentum spread, the value A p  is iiiomenturn half-width in that case. Jf the source is large, then, a roygh estimate shot~s N" = MH Sl, where Sl is the source width and MH the magnifiaation in the median plane, or horizontal plsne. cordingly one would expect, - Z A P = % M H S ~ / D , R  -r,, where D,/WB~ SIP and R being known parameters, one can evaluate A p  easily because a e  central my momentum can be known from a calibration of the spectrameter for its linearity4!%a@ the knowledge of magnetic rigidity curve. A C K N O W L E D G E M E N T S  ,'ghe author is thankful to Dr. Harald. A. Enge, Bead of the Righ Voltage. Engin'eer- ' k g -  Beptt NIT, and Dr. Kai Seigbahn, Prof. of Nuolar Physics, Nobel Iwtitutipn fur Fysikj Upsala, Sweden, for oommunim%ing their views and suggestions during the " "ogress of this work. The aufior has benefitted himself from time to t i ~ e  by personal " Xiscumions with Dr. Y. M. Bhise. .) REFERENCES I .  NIELSEN, 0. B. & KOFORD-HANSEN, O., Mat Fis Medd, Dan Vid Selsk, 29, No.6 (1955). 2. BAINBRIDGE, K. T., "Experimental Nuclear Physics I", (John Wiley & Sons, Inc New York), (19531,559. 3. JUDD, D. L., Rev Sci Insbr, 21, No.3 (1950),213. 4. STURROCK, P. A., "State & Dynamic Elictron Optics", (Cambridge University Press, Cambridge), (1952). 5. SIEGBAHN, K., "Alpha-beta-and gamma-ray spectroscopy Vol I", (North Holland Publishing Co Amsterdam), (1 965), 121. , 6 .  LAVATELLI, L., FEYNMAN, R.P. & SUTTON, R. B., "Snouting and Focusing the Cyclotron Beam", (52433, US Dep of Corn 0. Techn Serv), 1946, MDDC Report 350. 7. FOWLER, C.M., SHREFFER, R.G. & CORK, J.M., Rev SciImtr, 20, No.10 (1949), 966. 8. ROBINSON, C. F., Rev Sci Insfr, 20, No. 10 (19491,745. 9. ENGE, H.A., Rev Sci Instr, 35, No.3 (1 964), 278-287. 

Geometrical Ray Tracing of Ion-Trajectories for an Astigmatic Sector Focused Beta Ray Spectrograph

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Geometrical Ray Tracing of Ion-Trajectories for an Astigmatic Sector Focused Beta Ray Spectrograph

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