technique for 1st order design of a large-acceptance magnetic spectrometer

Nuclear Instruments and Methods in Physics Research A 481 (2002) 48–56

Technique for 1st order design of a large-acceptancemagnetic spectrometer

A. Cunsoloa,b,*, F. Cappuzzelloa,c, A. Fotib,d, A. Lazzaroa,b, A.L. Melitaa,b,C. Nociforoa,b, V. Shchepunove, J.S. Winfielda

a INFN - Laboratorio Nazionale del Sud, Via S. Sofia 44, 95123 Catania, ItalybDipartimento di Fisica, Universit "aa di Catania, Corso Italia 57, 95129 Catania, Italy

cCSFNSM, Corso Italia 57, 95129 Catania, Italyd INFN - sezione di Catania, Corso Italia 57, 95129 Catania, Italy

eFlerov Laboratory, Joint Institute of Nuclear Research, 141980 Dubna, Russia

Received 13 February 2001; received in revised form 19 April 2001; accepted 30 May 2001

Abstract

A general scheme of the layout of a large acceptance magnetic spectrometer based on a wide aperture quadrupole and

a multipurpose bending magnet is described. Physical quantities such as momentum resolution, focal plane size andinclination are explicitly represented as functions of transport matrix elements. In this way such quantities are directlyrelated to the parameters defining the configuration of the spectrometer. Realistic assumptions on the shapes, the

distances and the fields of the magnetic elements are taken into account in order to limit the parameter space to bespanned. A self-consistent technique simplifies the search for the best configuration and avoids the problem of ending inlocal minima. This technique is applied to the MAGNEX spectrometer, for which two competitive configurations,characterised by different bending angle, are found and discussed. r 2002 Elsevier Science B.V. All rights reserved.

PACS: 29.30

Keywords: Magnetic spectrometers

1. Introduction

The advent of facilities for the production ofradioactive ion beams has given rise in the last fewyears to a new interest in large acceptancemagnetic spectrometers as high-performance de-

vices for the detection of nuclear reaction pro-ducts. Specific interest arises from the excellentenergy and mass resolving power, the effectivesuppression of background and the possibility todetect charged particles emitted at very forwardangles, attributes demonstrated by magneticspectrometers in the past. Furthermore, ithas been recently shown [1,2] that such pro-perties, typical for small acceptance devices,can be at least partially maintained also formuch larger acceptance. This is a consequence

*Corresponding author. INFN - Laboratorio Nazionale del

Sud, Via S. Sofia 44, 95123 Catania, Italy. Tel.: +39-95-542-

328; fax: +39-95-714-1815.

E-mail address: [email protected] (A. Cunsolo).

0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 3 5 7 - 2

of the effective compensation, not possible in thepast, of the enhanced effects of aberrationproduced by large acceptance elements. Thekey point that has enabled such a breakthroughis the interplay of innovative designs of magneticelements and new techniques for the reconstruc-tion of particle trajectories through the spectro-meter.In general, the aberrations can be represented by

the second- and higher-order matrix elementsobtained by the Taylor expansion of final co-ordinates of the particles as function of the initialones [3,4]. A magnetic field with polarity 2ðnþ 1Þaffects only the values of the nth- and higher-orderaberrations. Thus, using convenient sets of mag-netic elements, each one designed to reduce theeffect of a strong aberration, it is possible, inprinciple, to get a satisfactory reconstruction ofthe image on the focal plane. This strategy is quiteinconvenient for large acceptance spectrometersmainly for the following reasons. Firstly, oneneeds to use a large number of magnetic elements,in order to correct the large number of non-negligible high-order aberrations. Secondly, alarge beam envelope within a correcting elementresults in an increase of the magnet cross-section,giving as a consequence rather encumbering andexpensive devices. Furthermore, an increased ratiobetween the cross-section and the length of amagnet will give a strong contribution to aberra-tions from the more extended fringing fields. As aconsequence the hardware corrections must berestricted to only a few parameters, principallyconnected with the design of the bending magnet(curvature of boundaries, introduction of fieldgradients between dipole pole pieces, etc.) ratherthan the introduction of separate multipolemagnets.A sizeable improvement in the correction of

aberration is obtained by using special algorithmsto reconstruct the relations between the initialphase space coordinates of particles with thosemeasured at the position of the focal plane. Bycalculating the transfer matrices and invertingthem, it is possible to reconstruct the initialcoordinates of particles from the measurement ofthe final ones by a focal plane detector. One shouldconsider that, due to the finite resolution of

detectors, it is not possible to completely compen-sate the residual aberrations by software techni-ques [3,5], therefore an effective reduction of theaberration strength by hardware design turns outto be fundamental.In this paper, we present a method for the first-

order design of large acceptance magnetic spectro-meters that was applied successfully to define theoptical configuration of the MAGNEX spectro-meter [6]. The latter consists of two magneticelements, a quadrupole and a multipurpose bend-ing magnet, the former being dedicated to providefocusing strength in the vertical plane and thelatter both momentum dispersion and horizontalfocus by rotation of the entrance and exitboundaries. One should emphasise that what-ever affects the first-order matrix elements (i.e.bending angle, boundary rotation, etc.) mayinduce higher-order aberrations. Details of thecompensation of the aberrations are not discussedin here, but will be presented in a forthcomingpaper [7].We will use the tensor formalism of Ref. [4],

according to which Rij ; Tijk and Dijkl define thefirst-, second- and third-order matrix elements,respectively. For example, T126 � ð@2xf =@yidiÞ;where the derivative is calculated along thereference trajectory.

2. Definition of the spectrometer layout

The requirement of large acceptance stronglyconstrains the design of a magnetic spectrometer.Obviously, one obtains an increased acceptance ifthe first magnetic element (quadrupole) is putcloser to the target and/or larger radii and gaps areused for magnetic elements. However, severelimitations arise from the manufacture of theelements, from space limitations and from thedisturbing effects on the optics of higher-orderaberrations, produced by huge magnets. All thesecombine to reduce the useful phase space, i.e. therange of detected angles and momenta with therequired resolution. Therefore, techniques forthe compensation of these unwanted effectsbecome a crucial point if these limitations are tobe overcome.

A. Cunsolo et al. / Nuclear Instruments and Methods in Physics Research A 481 (2002) 48–56 49

For a small acceptance magnetic spectrometerthere exists a close relationship between first- andsecond-order matrix elements and physical quan-tities as momentum resolving power, focal planelength and height, range of final angles and so on.Such relations become more complicated for largeacceptance spectrometers and additional, higher-order terms are needed. On the other hand, thefirst-order resolving power remains an importantparameter to maximise in the design of a spectro-meter, and, once optimised, provides a fasterconvergence for the higher-order terms in thesoftware correction. The maximisation of the first-order resolving power is one basic point of ourtechnique. Beyond this, in order to get an effectivecontrol of the physical quantities without losingthe advantages of the transport matrix technique,we use analytical formulae including higher-ordermatrix elements. In addition, we substitute thetraditional fit procedures of ray-tracing programs,where the constraints are the desired values ofcertain matrix elements, by a direct analysis of thetrend of the functions representing physicalquantities.In the following discussions all the matrix

elements referring to a plane normal to thecentral trajectory will be indicated by prime, todistinguish from those calculated in a planerotated by an angle C with respect to the planenormal to the central trajectory. The first step ofour technique is to find approximated expressionsfor the above functions in the plane rotated by anangle C; using appropriate coordinate transforma-tions, under the condition of point-to-pointimaging (R0

12 ¼ 0 and R034 ¼ 0). Neglecting the

horizontal dimensions of the source, the followingexpressions to third-order for the observablesxf ; yf ; and yf in the focal plane can easily befound:

xf ¼ 3R16dþ 12T122y

2 þ T126ydþ 12T166d

2

þ 12T133y

2 þ 12T144f

2 þ T134yfþ 16D1222y

3

yf ¼ 3Cþ R22yþ R26dþ 12T222y

2 þ 12T266d

2

þ T226ydþ 12T233y

2 þ 12T244f

2 þ T234yf

þ 16D2222y

3 ð1Þ

yf ¼ 3R 33yþ R34fþ T323yyþ T324yfþ T336yd

þ T346fdþ 16D3333y

3 þ 16D3444f

3

where the variables (y; f; d) are, respectively, theinitial horizontal and vertical angle and therelative deviation of particle momentum, and Cis the angle between the normal to the focal planeand the central trajectory. The symbol ¼3 indi-cates an approximation to third order. For aparticular value of C; given in Eq. (4), the T126matrix element vanishes. In the following calcula-tions, we refer to this case.Once the xf ; yf ; yf are calculated, it is possible to

reconstruct the focal plane length (lFP) and height(hFP) and the maximum horizontal angle at thefocal plane (ymaxf ) by

lFP � maxðxf Þ �minðxf Þ

hFP � maxðyf Þ �minðyf Þ

ymaxf � maxðyf Þ: ð2Þ

The momentum resolution depends on the termsaffecting the image size of a monochromatic beamon the focal plane. In the case of the MAGNEXspectrometer, it was chosen to measure theobservables (xf ; yf ; yf ; f), so the momentumresolution will depend on the derivatives(@xf=@yf ), (@xf=@yf ), (@xf=@f) which are compli-cated functions of matrix elements. By somealgebra we calculated the values of thesederivatives to third-order and transformed theresults onto the rotated focal plane. In thecalculations, the terms describing the mixingbetween horizontal and vertical motion of theparticles have been neglected. The relationsobtained are:

@xf@yf

¼ 3ð2T122yþD1222y

2Þ

ð2R22 þ 2T222yþ 2T226dþD2222y2Þ

@xf@yf

¼ 32T134f

2R33 þ 2T323yþ 2T336dþD3333y2

@xf@fi

¼ 3 �T134ð2R34 þ 2T324y þ 2T346d þ D3444f

2Þf2R33 þ 2T323yþ 2T336dþD3333y2

:

ð3Þ

A. Cunsolo et al. / Nuclear Instruments and Methods in Physics Research A 481 (2002) 48–5650

A check of Eq. (3), based on the Monte-Carlosimulations, has shown, within 30%, the reliabilityof these equations in a broad region of the phasespace around the reference trajectory. By calculat-ing the maximum values of these functions, andsumming them in quadrature, we are able toestimate the lowest limit of the momentumresolution.One should also consider that the matrix

elements directly depend on the parameters defin-ing the field geometry, the shape of the magneticelements and the distances between them. Forexample, the first-order matrix elements depend onthe following parameters: rQ; LQ and BQ (quadru-pole radius, length and field at the pole tip), o�

and oþ (entrance and exit face inclination angle ofthe dipole), rD and g (the dipole radius andbending angle), dtQ; dQD and dDF (distancesbetween target, quadrupole, dipole and focalplane), and a (first-order index field). In principle,the matrix elements can be set to desired valuesby manually adjusting the spectrometer con-figuration. In practice, due to the large numberof parameters involved, some of which act onsimilar matrix elements, a careful evaluation oftheir effect on spectrometer performance isnecessary.The next part of our procedure involves redu-

cing the parameter space by finding physical orother constraints, before detailed optical calcula-tions are performed. The use of the a parameterhas been left for a later step, to compensate for thekinematic effect, and is fixed to zero in the first-order design. Also, the point-to-point focus con-dition (having a coincident horizontal and verticalfocus) partially limits the phase space. In eachoptics calculation, for which we used the ZGOUBIray-tracing program [8], only one parameter ismanually changed in a grid of values spaced atfairly large intervals, while all the other parametersare frozen. The resulting matrix elements are putin Eqs. (1) and (3) and the results are used tocalculate the physical quantities of Eq. (2) and thethird-order resolving power, respectively. Thesemi-automatic grid search procedure has theadvantage that it reduces the chance of endingat a local minimum in the multi-parameteroptimisation.

3. Initial constraints

3.1. Bending angle

A large focal plane angle, which is defined as theangle between the normal to the focal plane andthe central trajectory, is undesirable because ofcomplications of detector construction and per-formance, and of the reconstruction software. Thefocal plane angle is a direct consequence of thehorizontal chromatic aberration. In the planenormal to the central trajectory the latter isrepresented by the second-order matrix elementT 0126 [4]. The value of this aberration is zero in a

plane whose normal is rotated by

C ¼ �arctanT 0126

R022R

016

ð4Þ

compared with the central trajectory, where R022

and R016 are determined in the plane normal to the

central trajectory.An initial set of calculations showed that it was

not possible to reduce T 0126 with acceptable values

of the parameters described above, mainly becauseof the contribution coming from the quadrupole.Instead, the behaviour of the product R0

22 R016 was

studied. From the Brown (R052 ¼ �R0

16=R011) [9]

and Liouville (R011R

022 ¼ 1) theorems for a focusing

device, expressed as first-order relations, oneobtains R0

52 ¼ R022R

016: This implies that a smaller

focal plane angle is expected for a larger R052

matrix element. In particular, Brown’s theoremshows that this element is directly proportional tothe first-order momentum resolution, with theconsequence that high first-order momentumresolution is an important requirement for thedesign of a large acceptance spectrometer. On theother hand, R0

52 also determines the path lengthdifferences between a fixed trajectory at a given yand the central one, which is a key factor for thereliability of the time of flight technique toseparate particles with different masses. Hence alarger R0

52 gives a smaller time of flight resolution,and consequently a poorer mass resolution for thespectrometer. So a compromise must be set at theinitial state of spectrometer design between mo-mentum resolution and mass resolution by time of


flight, taking into account also the undesirabilityof a large focal plane angle.Calculations show that R0

52 increases with thedipole bending angle [10] while T 0

126 remainsalmost constant, so smaller focal plane anglesare anticipated for dipoles with larger bendingangles. However, one should be careful thatthe matrix element R0

22 does not increase toomuch, otherwise the reduction of C may beannulled by an increase in the angular magnifica-tion. That can be avoided by keeping ymaxf to aminimum.A preliminary set of calculations for different

values of g in the interval gA½351; 801 showedthat the smallest focal plane angles are achievedfor the larger bending angles, as expected.Moreover, a further analysis of the momentumresolutions of Eq. (3) gave indications oftolerable values of the derivatives at g ¼ 551 and751: In the remaining part of this paper, weconcentrate on the optimisation procedure forthese two cases.

3.2. Dipole radius

The dipole mean radius, rD; determines themaximum magnetic rigidity or bending power ofthe spectrometer for a given BD: For MAGNEX,we wanted to accept reaction products withmagnetic rigidities of up to 1.8 Tm, which shouldbe sufficient for experiments with the Tandembeams, as well as for some light ion beams fromthe K800 cyclotron at LNS Catania. At the sametime, the dipole magnetic field should not exceedabout 1.2 T, to avoid problems of saturation. Thislimit is set lower than the usual level because of thealready large size of the return yoke of magneticelement. The maximum field indicates a value ofrDB1:5m. There are also considerations for howsmall or large the radius of the dipole can be basedon the required acceptance and cost of construc-tion. As regards ion optics, the radius affects theresolving power of the spectrometer and themaximum angle and size of the image at the focalplane. Bearing these factors in mind, a set ofcalculations was performed with rD steppedbetween the limits of 1.4 and 1.8m. The resultsshowed that while ymaxf decreased with increasing

rD; this was countered by an undesirable increasein lFP and hFP: The uncorrected third-order resolving power decreased slightly forrD larger than approximately 1.65m. Weconcluded that a reasonable comprise was to setrD to 1.6m.

3.3. Target–quadrupole distance

As stated above, the large acceptance conditionis achieved more easily if the distance between thequadrupole and the target is reduced. The relationis not as obvious as it might seem at the first sight,since the angular acceptance in the diverging planeis restricted by the beam envelope size at the exit ofthe quadrupole, and therefore is a function of theaperture and length of the quadrupole and, morecomplicatedly, the field shape and strength.Furthermore, in order to determine the mass ofthe detected particles, one needs a detector close tothe target that provides a start signal for a time offlight measurement. Also, it has been demon-strated [6,11] that the detector should also be ableto measure the scattering angle of particle in orderto improve the effect of the software techniques. Asuitable choice for such a detector is the m-channelplate with two-dimensional position readout de-scribed by Odland et al. [12]. This detectordepends on the focusing of electrons on a wire-matrix which would be disturbed by the fringingfield of a large magnetic element. To have anacceptable measurement of the angle, the detectorshould be about 20 cm from the target. Hence thefringe field of the quadrupole must commencebeyond this distance. However, the fringing fieldof the lens extends a distance of the order of itsdiameter. Thus an increase in the radius ofquadrupole forces an increase of the distance ofthe magnet away from the detector, with dimin-ished gain in the acceptance. Initial optics calcula-tions showed that an acceptable compromise forMAGNEX is obtained with a 20-cm radius,60-cm long quadrupole place at 40 cm from thedetector. This choice also gives a suitable focaldistance, an acceptable pole-tip field (1 T) for thequadrupole and a tolerable value of the T346aberration.


4. Optimisation procedure for a bending

angle of 551

In this section, all the discussions refer tocalculations with the dipole bending angle g fixedto 551. Also, the dipole radius was set to 1.6m, thequadrupole parameters fixed as determined inSection 3.3 and a set to zero. The optimisationprocedure was divided into three steps.

4.1. Optimisation by inclination of dipole entranceand exit faces

In this step, the behaviour of the approximated‘‘physical’’ parameters mentioned above, namelylFP; hFP; y

maxf ; the resolving power andC; has been

studied in a grid of values of o� and oþ: In eachcalculation, one parameter is manually changedwhile the other is frozen. The parameters BQ; dQD;and dDF are deduced from a best fit to the point-to-point focus condition (R0

12 ¼ R034 ¼ 0). Then the

second- and third-order matrix elements areobtained and transformed into the rotated focalplane reference frame. Finally, the physical quan-tities are calculated from Eqs. (2)–(4).

The calculations are divided into 20 groups.Each group has a fixed value of o� in the rangefrom �301 to 81 with a step of 21, while oþ variesin order to obtain the double-focus condition. Inthe interval of o�A½�221; � 161 the highestthird-order resolving power is obtained togetherwith reasonable values for the dimensions of focalplane, as shown in Figs. 1 and 2. For o� ¼ �181the first-order resolving power weakly depends onoþ (see Fig. 1). For o� ¼ �161 a smaller value ofymaxf could be obtained than for o� ¼ �181(Fig. 3), but for the former, the first-order resol-ving power fluctuates rapidly with oþ: For thesereasons we chose o� ¼ �181: Fixing the value ofo� at this value and tuning on oþ; a plateau in thethird-order resolution (Fig. 1) and a minimumdimension of the focal plane (Fig. 2) are observedaround oþ ¼ �181; which is taken as the finalresult.

4.2. Quadrupole–dipole separation

After fixing the values of o� and oþ the effect ofdQD parameter on the physical quantities wasstudied in a grid of values between 100 and 135 cm.In the calculations the parameters dDF and BQ are

Fig. 1. The first-order resolving power and the lowest limit of the uncorrected third-order resolving power for four different values of

o� (~� 221; ’� 201; m� 181; �161). These curves were obtained in the first step of the optimisation procedure. They are used toguide the choice of best o� and oþ values.


determined by the point-to-point focus conditionsR012 ¼ R0

34 ¼ 0:In the design of a high-resolution magnetic

spectrometer the distance between magnetic ele-ments should exceed the sum of fringe field

extensions to avoid undesirable overlaps of thefield. The extension is of the order of the diametersize for a quadrupole and two times the verticalgap for a dipole. For large acceptance elementsthis amounts to several tens of centimetres (e.g.B80 cm for MAGNEX).The dQD parameter has also a critical impact on

the efficiency of the MAGNEX spectrometerbecause of the large acceptance. The beamenvelope after the single-lens quadrupole reducesvertically while enlarges horizontally. Therefore,the shorter this distance is, the bigger must be thedipole gap to accept the beam without losses. Forthe same reason the longer dQD is, the larger mustbe the dipole horizontal size. From a construc-tional point of view, bending magnets with gapslarger than 20 cm or horizontal size larger than150 cm are hardly feasible. Thus a compromisebetween all these factors must be found at thisstage. The calculations indicate that for MAG-NEX the solution dQD¼ 130 cm represents anacceptable solution.

4.3. Iteration

In this step the procedure described in Section4.1 is repeated, but starting from the optimisedvalues of the distance dQD in order to obtain a

Fig. 2. Horizontal (left) and vertical (right) focal plane length for four different values of o� (~� 221; ’� 201; m� 181; �161).These curves were obtained in the first step of the optimisation procedure and are used to guide the choice of best o� and oþ values.

Fig. 3. Curves of the largest horizontal angle in the focal plane

inclined by c obtained in the first step of the optimisation for

four different values of o� (~� 221; ’� 201; m� 181;� 161). These curves were obtained in the first step of the

optimisation procedure and are used to guide the choice of best

o� and oþ values.


more precise definition of o� and oþ: After that,the new values of the pole faces angles are insertedin the calculations of Section 4.2, and a new valueof dQD is obtained. This self-consistency techniquegoes on until the result of the calculations givesvalues differing from those of the previous itera-tion by a negligible amount. In the case discussedhere for the 551 dipole of MAGNEX, twoiterations are enough to achieve convergence.

5. Results and discussions

The self-consistent technique discussed abovemade it possible to find an optimised first-orderconfiguration of the spectrometer for a bendingangle of 551. This is characterised by a minimumenlargement of image size and a significantreduction of the maximum horizontal angle onthe focal plane. In particular, the value of focalplane angle of the optimised configuration isC ¼ 611: This solution also gives an acceptablesize of the focal plane detector.It should be emphasised that up to this point we

have been dealing with the uncorrected third-orderresolving power, lFP; hFP; and ymaxf : These approxi-mated values are useful in the determination of thefirst-order layout, but should not be taken as arepresentation of the final values. Once the first-order layout is determined, a partial compensationof the residual aberrations is obtained by appro-priate curvature of the dipole boundaries. Duringthis correction procedure, not only does the high-order resolving power increases, but also lFP andymaxf are significantly reduced. Unfortunately, thisis at a cost of an increase of hFP: The values ofthese parameters after optimisation of the curva-ture of the dipole boundaries are given in Table 1.A partial compensation of the kinematic effect isachieved by quadrupolar and hexapolar fieldsgenerated by first (a)- and second (b)-order surfacecorrecting coils [13]. Again in this case, thetechnique has as its goal the minimisation of thederivatives of Eq. (3), rather than the optimisationof simple matrix elements. Further improvementsin the momentum resolution are obtained by theapplication of ray reconstruction algorithms. Afinal resolution of about 2000 can be achieved for

the MAGNEX spectrometer throughout a broadangular (B50msr) and momentum acceptance(710%). Details may be found in a forthcomingpaper [7].All the procedure discussed in Section 4 has

been secondly applied to the case of a bendingangle of 751. The results, after compensation ofaberrations and kinematic effect, are comparedwith those for a bending angle of 551 in Table 1.The configuration with the bending angle of 751

presents an appreciable decrease in the focal planeangle which is an important advantage since itmakes easier the operation of the focal planedetector as well as reducing the straggling pro-duced by the entrance window of the focal planedetector. Moreover, the first-order resolution ishigher because of the enhanced dispersion. How-ever, after the optimisation of the dipole boundaryprofiles, a high value of the derivative (@xf=@f) isobserved in a significant part of the phase spaceaccepted by the spectrometer. As a consequence,the requirement on detector performances for aneffective reconstruction of the trajectories becomestoo stringent for part of the total phase space anda smaller usable acceptance is obtained (about46msr).

Table 1

Optical characteristics of MAGNEX spectrometer optimised

for bending angles of 551 and 751. The values for the focal plane

angle, length and height, and for ymaxf are the results after

correction by shaping the dipole boundary profiles

Optical quantities g ¼ 551 g ¼ 751

Maximum magnetic rigidity (Tm) 1.8 1.8

Solid angle (msr) 52 46

Horizontal angular acceptance (mr) �90, +110 �90, +110

Vertical angular acceptance (mr) 7130 7130

Momentum acceptance 710% 710%

Mean radius (cm) 160 160

Maximum radius (cm) 255 255

Minimum radius (cm) 85 85

Central path length (cm) 592 616

Momentum dispersion (cm/%) 3.98 4.51

First-order resolution 5000 7000

Focal plane angle 611 541

ymaxf ðd ¼ �0:1Þ 74.51 63.91

ymaxf ðd ¼ 0:1Þ 77.71 70.01

Focal plane length (cm) 92 100

Focal plane height (cm) 32 32


Furthermore, the 751 bending angle solutionresults in a massive dipole (B120 t), which wouldneed extraordinary manufacturing procedures andwould create severe problems for the supportingplatform, for instance. As a result such a magnetwould be considerably more expensive than the551-bend one.

6. Summary

The layout of the large acceptance QD magneticspectrometer MAGNEX has been defined. Theparameters of the configuration are refined withina new approach based on the optimisation ofphysical quantities of interest described as functionof transport matrix elements and a self-consistentmethod for exploring the phase space. Twoconfigurations, characterised by different bendangles, are compared. That at 551 offers the bestcompromise between the requirements of ionoptics and the manufacturing of the magneticelements and the associated utilities.

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technique for 1st order design of a large-acceptance magnetic spectrometer

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