Ion optics for large-acceptance magnetic spectrometers: application to the MAGNEX spectrometer

Nuclear Instruments and Methods in Physics Research A 484 (2002) 56–83

Ion optics for large-acceptance magnetic spectrometers: application to the MAGNEX spectrometer A. Cunsoloa,b,*, F. Cappuzzelloa,c, A. Fotib,d, A. Lazzaroa,b, A.L. Melitaa,b, C. Nociforoa,b, V. Shchepunova,e, J.S. Winfielda b

a INFN, Laboratori Nazionali del Sud, Via S. Sofia 44, 95123 Catania, Italy Dipartimento di Fisica e Astronomia, Universita" di Catania, Corso Italia 57, 95129 Catania, Italy c CSFNSM, Corso Italia 57, 95129 Catania, Italy d INFN, Sezione di Catania, Corso Italia 57, 95129 Catania, Italy e Flerov Laboratory, Joint Institute of Nuclear Research, 141980 Dubna, Russia

Received 10 April 2001; accepted 13 September 2001

Abstract The ion optics of large-acceptance magnetic spectrometers are discussed. General techniques based on a minimum of multi-purpose magnetic elements are described. The aberrations should be minimised by shaping the entrance and exit effective field boundaries of bending magnets, the residual terms being corrected by software. Field clamps, shims and surface coils (the latter to provide kinematic compensation) are also discussed. The results and formulae which we obtain are applied to the case of the large-acceptance (B50 msr) high-resolution magnetic spectrometer ‘‘MAGNEX’’ at INFN-LNS Catania. r 2002 Published by Elsevier Science B.V. PACS: 29.30.
h; 29.30.Aj Keywords: Charged-particle magnetic spectrometers; Ion optics

1. Introduction With many radioactive nuclear beam facilities operating or under construction throughout the world, considerable effort is being directed to new instrumentation designed on the basis of the specific properties of secondary beams. In particular, the relatively low intensities compared to stable beams (at least three orders of magnitude lower) invoke a need for large acceptance devices. While in several laboratories, large segmented arrays of, e.g., silicon detectors are used or planned, the superior energy and mass resolution of a magnetic spectrometer, together with its clean separation of unwanted ion species, makes such devices an attractive choice for charged-particle spectroscopy. Thus it is pertinent to consider the ion optics of magnetic spectrometers beyond the limits of ones in present use. In the case of weak intensity beams, a large *Corresponding author. INFN, Laboratori Nazionali 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 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 2 0 0 4 - 6

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acceptance is not only important in terms of overall counting rate, but also means that angular distributions can be measured with relatively few angle settings. At the same time, one wishes to approach the highresolving power achieved by smaller spectrometers, since this is both a partial compensation for lowstatistics spectra (improving the signal to noise ratio) as well as an advantage in the separation of possible close-lying states. The two criteria of large solid angle of acceptance together with high resolving power mean that the ion-optical aberrations must be very carefully treated. In this paper we examine the consequences of magnetic spectrometry at large acceptance, and demonstrate that many of the traditional solutions for high-resolution devices (see, e.g. Ref. [1]) cannot be applied when the approximations and truncations of the small acceptance limit are no longer valid. In Section 2 we discuss in general terms the main challenges encountered in the design of a largeacceptance magnetic spectrometer, which are chiefly induced by strong aberrations and by the kinematical effect. In Section 3 our formulae and conclusions are applied to a new magnetic spectrometer ‘‘MAGNEX’’ which has been designed primarily for the radioactive beam facility EXCYT [2] at INFN-LNS Catania.

2. Large acceptance spectrometers 2.1. Reference frame The usual way to describe the motion of particles in a beam is based on the choice of the motion of one of them as a reference (see, e.g., Ref. [3]). Its path through the magnetic elements is called the reference trajectory and its momentum the reference momentum. The reference momentum p0 is used to set the magnetic strength of the bending magnets (dipoles). The positions and momenta of the other particles may be defined relative to the reference ones. At any point along the reference trajectory, we define a longitudinal or t-axis in the direction of the reference momentum. The two transverse axes x and y are usually chosen perpendicular to it. To specify the momentum of a particle three quantities are used: px ; py ; the momentum components along the x and y directions and d ¼ ðpFp0 Þ=p0 ; the fractional deviation from the reference momentum. One also considers the quantities x0 ¼ px =pt and y0 ¼ py =pt ; where the pt is the momentum longitudinal component along reference trajectory. Because px and py are small compared to pt ; x0 and y0 can be approximated by the horizontal y and vertical f angles with respect to the reference trajectory.1 Three other parameters are needed in the coordinate set: x and y; the two transverse distances of particles from the central trajectory, and l, the difference in path length between a given trajectory and the reference one. The path length is defined as the distance along the particle trajectory, between the starting point of the beam line (the target position for magnetic spectrometry) and the intersection between the trajectory and a plane normal to the central trajectory, at the fixed t: In summary, each particle will be characterised by the following set of observables: x; y; y; f; l and d: A scheme of a generic optical reference frame with the first four quantities defined above is shown in Fig. 1. 2.2. Matrix formalism and aberrations In the phase space representation introduced in the previous section, the motion of a charged particle beam, under the action of magnetic fields, can be represented as the dynamical evolution of the representative hyper-volume. In particular, the final position Pf  ðxf ; yf ; yf ; ff ; lf ; df Þ in such a space 1

In this paper we assume x horizontal and y vertical.

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r φ

px

py

p θ

pt

M

x

y

t reference trajectory

M'

Fig. 1. Example of right-handed coordinate system for charged particle optics.

is connected to the initial Pi  ðxi ; yi ; yi ; fi ; li ; di Þ by a general relation as follows: F : Pi -Pf :

ð1Þ

Eq. (1) describes a general non-linear transport relation, characteristic of the particular optical system. More explicitly one obtains xf ¼ F1 ðxi ; yi ; yi ; fi ; li ; di Þ yf ¼ F2 ðxi ; yi ; yi ; fi ; li ; di Þ yf ¼ F3 ðxi ; yi ; yi ; fi ; li ; di Þ ff ¼ F4 ðxi ; yi ; yi ; fi ; li ; di Þ lf ¼ F5 ðxi ; yi ; yi ; fi ; li ; di Þ df ¼ di

ð2Þ

where the last equality expresses the conservation of momentum modulus in purely magnetic fields in the absence of degrading materials. The parameter li is essentially constant for our case of thin targets, and will be omitted from the discussion in this paper. Rewriting Eq. (2) in tensor notation, one has X X Rjk xk ðiÞ þ Tjkl xk ðiÞxl ðiÞ þ ? ð3Þ xj ðf Þ ¼ k

k;l

where xj is the generic phase space coordinate, Rjk and Tjkl are, respectively, the first and second order transfer matrix elements. The T matrix elements are second partial derivatives, calculated close to the reference motion, such that the T tensor is symmetric to the exchange of two indexes. The main advantage of the tensor formalism is that the R and T tensors, for a complex magnetic optical system, can be constructed as a product of the analogue tensors of the single magnetic elements [3]. The solution of this system is generally obtained by means of numerical algorithms. In the limit of small deviations from the reference trajectory and momentum, the Taylor expansion of all the Eqs. (3) can be truncated to first order. However, as will be demonstrated in the following sections, a

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first order truncation is not valid for a large-acceptance magnetic spectrometer. Instead, strong effects arise from second, third and higher order coefficients of the Taylor expansion. Since these terms worsen the system performance compared to that expected by first order calculations, they are often referred to as aberrations. In general, the compensation of the aberrations at a certain order can be obtained by means of magnetic elements that modify the matrix elements at that order, without changing those at lower order. In any case, it is not a trivial task to find the relationship between coefficients of Taylor expansion and physical quantities connected with the shape of magnetic elements. An exact description for such a relationship is practical only for first and second order coefficients. 2.3. First order design In the design of a large-acceptance magnetic spectrometer, particular attention must be paid to making a realistic compromise between the momentum resolution Rp to be achieved and the length lFP and height hFP of the focal plane. To 1st order these quantities are bound by simple relations to R matrix elements lFP ¼1 R11 Dxi þ R12 Dyi þ R16 Dp hFP ¼1 R33 Dyi þ R34 Dfi R16 Rp ¼ 1 : R11 Dxi

ð4Þ

In the above formulae the ¼1 symbol indicates a first order approximation while the D represents the acceptance parameters of the spectrometer. For a double focusing system R12 and R34 must be coincidently zero and this reflects on the allowed magnetic strengths of focusing magnets (quadrupoles) and on the focal length. Furthermore, since each quadrupole field focuses in a given direction but defocuses in the normal to that direction, a requirement of double focus usually needs at least two quadrupoles. For a large-acceptance magnetic spectrometer the second quadrupole would need to have a very large radius to accept the beam envelope from the first, being therefore rather difficult to design and construct. Such a solution has been adopted for the VAMOS spectrometer [4] at GANIL where the second quadrupole is designed to have an elliptical shape with a 100 cm horizontal aperture. An alternative and simpler way to provide a vertical focus after a single horizontally focussing quadrupole is to put an extra inclination on the dipole boundaries compared to the ideal shape of circular sector [3]. If the beam spot size at the target is not large, the linear horizontal and vertical magnification terms (R11 Dxi and R33 Dyi ) in Eq. (4) contribute only a few millimetres to the focal plane size compared to the several tens of centimetres produced by the dispersive term in the horizontal plane and the higher order terms for the vertical one in case of a large momentum acceptance spectrometer. If R11 Dxi and R33 Dyi are then assumed to be negligible, the length lFP of the focal plane is essentially given by R16 Dp and the height hFP is essentially zero (in the first order approximation). Note also that both lFP and Rp depend linearly on the dispersion (R16 ). As a consequence, a demand for a high momentum resolution leads to a large focal plane and a compromise must be found for a large momentum acceptance. Moreover, the dispersion depends only on the dipole field shape, therefore such a compromise strongly influences the geometry of dipole magnets. 2.4. Aberration compensation From the discussion of the previous section it follows that to compensate unwanted 2nd order aberrations a number of sextupolar magnetic fields have to be put at appropriate positions in the spectrometer. A sextupolar magnetic field can be generated by means of a pure sextupole lens, or by

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parabolic shaping of the pole tips of dipole magnets or by designing their effective field boundary profiles in the form of circular arcs. For large acceptance spectrometers, magnetic multipole magnets would need to have a large radius of, say, between 50 and 100 cm. For such a magnet, the fringe field region extends for about twice the radius both externally and internally. Thus a length of more than twice the radius would have to be used in order that the fringe field is not dominant. Such large magnets are unpractical and expensive to build. One might also consider ‘‘dipolar corrector’’ magnets [5]. These may provide a solution in principle, but again it would be an expensive one. The number of sextupole fields to be introduced depends on the number of 2nd order aberrations to be compensated. Such aberrations are mainly generated by the large acceptance focusing elements of the spectrometer and affect principally the motion of particles at the border of the phase space. In particular, for a system with point-to-point focusing, the chromatic terms T116 ; T126 ; T336 and T346 mainly contribute to the image size. Other important aberrations for a spectrometer are the geometrical ones, such as T122 ; T144 ; D1222 ; D1244 and so on, Dijkl being the transfer matrix elements of 3rd order. In the case of ionbeam spectrometry, the terms associated with the beam spot size at the target, T116 and T336; have negligible effects for applications with, e.g., Tandem beams, while the chromatic terms T126 and T346 can be quite important. The compensation of T126 worsens the effects of T346 and vice versa. Consequently for large acceptance spectrometers, it not being feasible to have more than one sextupole pair, a compromise must be obtained. In practice T126 produces a rotation of the focal plane while T346 produces a momentum-dependent enlargement of the vertical dimension of the focal plane. A rotation by an angle c around the plane normal to the central trajectory at the focus position transforms the T126 as follows: c ¼ T126

T126 þ R22 R16 tan ðcÞ : cos ðcÞ

ð5Þ

Thus this aberration has no effect in a plane rotated by an angle given by T126 tan ðcÞ ¼
: R22 R16

ð6Þ

In the past, the T126 aberration was sometimes compensated by introducing an appropriate second order component (gradient) into the field of the dipole. This solution is not suitable for large acceptance spectrometers, because the gradient field introduces 2nd and higher order aberrations, such as T122 and T144 : These aberrations can only be successfully compensated for small acceptance, since they increase catastrophically with angle and momentum. An alternative technique is to determine directly or by reconstruction, the position of particles on the inclined focal plane. This requires a careful evaluation of the location of the position detectors (PD1 and PD2) relative to the focal plane, as discussed below. Particles passing through the material of the focal plane detector suffer small angle multiple scattering, producing a spread in the xf ; yf ; and yf measurements. This straggling Dystr obviously affects the extrapolation of the position of the focal plane. For an ‘‘orthogonal geometry’’, in which the detectors are normal to the central trajectory (see Fig. 2), the extrapolated position of the focal plane can be written as xf ¼

0:5ðx1 þ x2 Þ cos ðcÞ
d
1 sin ðcÞðx1
x2 Þ

ð7Þ

where x1 and x2 are the horizontal coordinate measured in PD1 and PD2, yf is the horizontal angle of the ion trajectory on the focal plane with respect to the central trajectory and d is the distance between the two planes on which the horizontal position is measured. If Dx1 and Dx2 are the intrinsic position resolutions of the two detectors, the square of the contribution of such a ballistic effect to the error in the focal plane

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Fig. 2. Sketch of the focal plane detector with position detectors orthogonal to the central trajectory (plan view).

horizontal position measurement is  2  2 0:5 cos ðcÞ
x2 d
1 sin ðcÞ 0:5 cos ðcÞ
x1 d
1 sin ðcÞ 2 2 4 Dxf ¼ cos ðyf ÞðDx1 Þ þ cos2 ðc þ yf Þ cos2 ðc þ yf Þ  2 x1 sin ðcÞ þ 0:5d cos ðcÞ  cos4 ðyf ÞðDx2 Þ2 þ ðDystr Þ2 : cos2 ðc þ yf Þ

ð8Þ

For large acceptance spectrometers the value of c may well reach values >601, which would result in a strong ballistic effect, both degrading the energy resolution and causing an unwanted dependence on yf and xf : The problem can be avoided by inclining PD1 such that it lies along the focal plane. Then one obtains s Dxf E Dyf ð9Þ cosðc þ yf Þ where s is the distance between the entrance window and PD1 measured along the particle trajectory. Eq. (9) shows that the xf measurement in this configuration is no longer affected by ballistic error. However, the inclination of the detector increases the effective width of the window by a factor of cos
1 ðc þ yf Þ: 2.5. Hardware methods to compensate aberrations The 2nd order aberration theory can be quite inaccurate for very large magnetic elements. Moreover, as a general rule, a system composed by N magnetic elements, each affected by order M intrinsic aberrations, can be affected by induced aberrations up to M N order. As discussed in Section 2.4, it is not convenient to use multipole magnets in such conditions. For this reason it is useful to develop specific techniques to compensate such higher order aberrations, based on a minimum of multi-purpose magnetic elements. One of these techniques, invented by Brown [6], consists in relating aberrations up to the eighth order to the coefficients of a polynomial profile of the dipole field boundaries. In particular, in the RAYTRACE program [7] the profile of the dipole boundaries are described by the function  i 8 Dt X x
¼ Si ð10Þ r r i¼2 where Dt corresponds to the deviation from a straight line profile and r is the central radius. The first order term is not considered since it describes the pole face rotation, which is already exploited as an extra focusing element. The Si are the Brown coefficients, which can be adjusted compensate a given aberration. For each boundary only one aberration can be corrected at each order and we have two profiles for each

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dipole (entrance and exit). The entrance field boundary does not introduce chromatic effects, while the second does, being situated in a dispersive region. One way to compensate the effects of higher order aberrations is to increase the number of dipoles. This has often been a solution chosen for small acceptance spectrometers in the past, most notably in the Q3D series of the 1970s [8]. One should bear in mind that the correction process must be aimed to optimise the image width, and not necessarily to compensate a particular aberration. Thus it can happen, especially in the extreme conditions of a large acceptance spectrometer, that the minimum width of the image is achieved in conditions where second order aberrations are not completely eliminated. In Section 3 a similar situation will be described for the MAGNEX spectrometer. 2.6. Kinematic effect The reaction products from a two-body process a þ A-b þ B have a momentum p that depends on the scattering angle yl (in the laboratory reference frame). If the spectrometer is positioned at an angle of y0 to the incident beam, expanding the d parameter to first order around y0 one obtains   qd dðyl Þ Edðy0 Þ7 ðyl
y0 Þ qyl y0 ¼ dðy0 Þ7kðy0 Þðyl
y0 Þ

ð11Þ

where k is the so called kinematic coefficient of the reaction2, and ðyl 2y0 Þ represents the scattering angle in the spectrometer reference frame which corresponds to the absolute value of the phase space y-parameter. The sign before the kðy0 Þy term is important, and should be clearly understood. First, one could potentially obtain a different sign of k depending on which kinematic solution is taken for inverse kinematic reactions. Here we only consider the usual first solution for the kinematics where the outgoing particle energy decreases with angle, hence k is negative. Secondly, the sign of y depends on the relative orientation of the optical reference frame and the laboratory (incident beam) reference frame, as shown in Fig. 3. The sign of these angles agrees if the spectrometer whose dipole bends to the right is placed on the left-side of the beam and vice versa. Applying energy and linear momentum conservation principles, one obtains the following nonrelativistic expression for k: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ma Mb Ea =Eb sinðy0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k¼
: ð12Þ Mb þ Ma
Ma Mb Ea =Eb cosðy0 Þ The magnitude of k increases with scattering angle and is dependent on the relative masses of the target and projectile. Because we now have an angular dependence of the momentum, the final position of a particle is transformed as follows: 0

xf ¼ F1 ðxi ; yi ; yi ; fi ; dÞ ¼ F1 ðxi ; yi ; yi ; fi ; dðy0 Þ7kyÞ;

ð13Þ

where the additional y-dependence induced by the chromatic matrix elements is shown. As a consequence the R12 element, which must be zero to achieve the focus condition on the horizontal plane, is transformed as R012 ¼

qxf qF1 qF1 ¼ 7k ¼ R12 7kR16 : qyi qyi qd0

ð14Þ

2 We define k as þ1=p dp=dy; which follows the convention of, e.g., Sugiyama et al. [9], but is the opposite sign to that used in, e.g., Enge [10].

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Fig. 3. Sign of y in Eq. (11) with respect to the orientation of the spectrometer relative to the (fixed) incident beam in the laboratory.

Differentiating this with respect to yi ; one obtains 0 T122 ¼ T122 72kT126 þ k2 T166 :

ð15Þ

Eqs. (14) and (15) show that the kinematic effect worsens the final horizontal position resolution owing both to the 1st order focus shift and to the enhancement of the spherical aberrations. Higher order effects arising from the dependence of p on y can be studied in similar manner. In particular, we note that Eq. (6) for the focal plane angle c now becomes T126 : tan ðcÞ ¼
R16 ðR22 7kR26 Þ

ð16Þ

It is important to note that the trajectory length is also affected by the k factor. The induced error on the reconstruction of L becomes DLE½R52 7kR56 Dy

ð17Þ

in which the effect of k is clear. A detailed study of this effect is given in Ref. [9]. The error on the trajectory length has direct consequences for the mass resolution, as will be shown later. One should also take into account the change in velocity of the particle with angle, so the effect of k on the mass resolution is not straightforward. The simplest way in hardware to reduce the 1st order effect of kinematical line broadening is to shift the focal plane detector to the position of the new focus. This shift is determined by the condition R012 ¼ 0 ) R12 ¼ 8kR16 :

ð18Þ

In our convention, if k is made more negative, the focal plane is shifted towards the dipole exit, and vice versa. The detector-shift procedure does not compensate the second and higher effects produced by the k value and is therefore only partially effective for large acceptance spectrometers. More refined

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techniques are needed in such cases. One possibility is to put the symmetry plane of the spectrometer normal to the scattering plane, by rotating the dipole by 901 into the vertical plane. An example of such a spectrometer is the superconducting S800 at MSU-NSCL [11]. By this means the scattering plane becomes the vertical plane in the spectrometer reference frame and the vertical component of the scattering angle is analysed in the dispersive plane. This component decreases with the scattering angle, tending towards zero at 901. On the other hand, at very forward angles, the component to the scattering angle coming from the vertical angle is comparable with the horizontal one, and the kinematic effect is no longer compensated. One should note that it is just these very forward angles that are important for strongly inverse kinematics reactions which are a common feature of radioactive beam experiments, and indeed measurements at forward angles are often critical in many direct nuclear reaction studies. Another possibility to compensate the first order kinematic effect is to change the focus strength by quadrupole magnet, as done for the RAIDEN spectrometer [12]. Higher order effects (curvature of the focal plane) can be compensated by multipole magnets, as also demonstrated for the Oxford MDM-2 spectrometer [13]. For large acceptance spectrometers the quadrupole-adjustment technique can only fully compensate for small k values, otherwise the beam envelope in the vertical plane becomes unacceptably large. Therefore the use of this technique requires a detailed analysis of acceptance requirements. For large acceptance spectrometers an attractive technique is the use of large thin coils, called surface coils, which are placed inside the dipole gap to provide a supplemental multipolar field [14]. This is a convenient option that avoids the expense of building large external magnets. Furthermore, the problem of transmission efficiency through the dipole is reduced, since the correction fields are now after the entrance boundary of the dipole and distributed along its length. Finally, we should mention that in some spectrometers (e.g., SPEG [15] at GANIL), the kinematic correction is done entirely by software reconstruction of the focal plane. This method relies on a sufficiently accurate determination of the trajectory angles in the focal plane, and is only feasible for spectrometers with focal planes nearly normal to the particle trajectories. 2.7. Limits of software aberration correction Hardware techniques do not always provide a satisfactory minimisation of all the important aberration terms. The increase in angular and momentum acceptances, which is the main feature of modern spectrometers, makes this minimisation process more critical. One must therefore consider correcting the effects of aberrations also by software procedures, which implies the use of measured angles. An important consideration is whether the angles should be measured immediately after the target (i.e. before the magnetic elements) or at the focal plane. The measurement immediately after the target is the most direct, but it will generate straggling effects from the intercepting detector. The straggling will interfere with the spectrometer optics, perhaps resulting in even worse performance than it is intended to correct! Hence it is often necessary to measure angles at the focal plane or, at least, after having passed through magnetic fields. Extensive mapping of the fields is then crucial to the overall performance of the spectrometer. On the other hand, an angular amplification resulting from the presence of a quadrupole field, helps to make the use of the angle at the focal plane more advantageous. The software correction technique is fundamentally limited by the accuracy in the angle measurements, as the following simple example makes evident. Let us suppose that the performance of a spectrometer is affected by a D1222 aberration. The consequence is that the xf coordinate of the particles with the same magnetic rigidity will be distributed over a range of x0f values given by x0f ¼ xf þ D1222 y3 ¼ xf þ C

ð19Þ

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where y is the angle at the target. If sy is the error in the measurement of y; the propagated error in the correction parameter C will be given by sC ¼

qC sy ¼ 3D1222 y2 sy : qy

ð20Þ

The ratio between the image broadening without and with angle measurement (the fractional error on the aberration coefficient) is 3sy =y In the case of large acceptance spectrometers this can be significant: for values of sy ¼ 10 mrad and y ¼ 100 mrad one finds that the D1222 aberration can only be corrected by software to the 30% level. The conclusion is that even if software corrections are to be applied, a largeacceptance spectrometer must be first designed to minimise by hardware the aberration effects. 2.8. Ray-reconstruction method The performance of a spectrometer can be represented by means of the transport operator F in the coupled Eqs. (2). The parameters to be measured in an experiment are the scattering angles and the momentum modulus at the target. It is necessary to invert the set of Eqs. (2) with respect to the initial set of parameters, obtaining xi ¼ F10 ðxf ; yf ; yf ; ff ; lf Þ yi ¼ F20 ðxf ; yf ; yf ; ff ; lf Þ yi ¼ F30 ðxf ; yf ; yf ; ff ; lf Þ fi ¼ F40 ðxf ; yf ; yf ; ff ; lf Þ d ¼ F50 ðxf ; yf ; yf ; ff ; lf Þ:

ð21Þ

In applications where it is convenient to measure some parameter directly after the target, Eqs. (21) will be substituted by a similar system expressed as a function of the measured quantity. For example, in the case of the MAGNEX spectrometer the chosen set of independent variables is given by ðxf ; yf ; yf ; fi ; xi Þ; so we invert Eqs. (2) to produce the following coupled set: yi ¼ G1 ðxf ; yf ; yf ; fi ; xi Þ yi ¼ G2 ðxf ; yf ; yf ; fi ; xi Þ ff ¼ G3 ðxf ; yf ; yf ; fi ; xi Þ lf ¼ G4 ðxf ; yf ; yf ; fi ; xi Þ d ¼ G5 ðxf ; yf ; yf ; fi ; xi Þ:

ð22Þ

By measuring the ðxf ; yf ; yf ; fi ; xi Þ quantities and knowing the G functions it will be possible to reconstruct the ðyi ; yi ; ff ; lf ; dÞ quantities. A powerful method to relate the initial coordinates to the final ones was developed in the MOTER code [16], which, once the magnetic elements of the spectrometer are specified, starts a ray-tracing procedure of a large number of trajectories spanning the whole phase space. The parameters of the G functions are obtained by a polynomial fit. Unfortunately this procedure needs a large computational effort, which limits the number of coefficients which can be calculated. However, subsequently a more rapid technique was developed [17]. This technique, based on the formalism of algebraic differences [18], makes possible the calculation of relatively high order matrix elements, and avoids lengthy ray-tracing procedures. This is done by use of the following recurrence formula: n Mn ¼n ðA
1 1 3 ðI
An 3Mn
1 ÞÞ

ð23Þ

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where the symbol ¼n means that the product is truncated to the nth order. The COSY INFINITY program [19] contains such an algorithm providing at the same time a detailed evaluation of magnetic fields. In this way the Mn matrix can be obtained by iteration with the only condition that the A1 determinant must be non-zero. It is possible to demonstrate that this condition is verified in the case of dispersive magnetic fields, for which the determinant is directly given by the 1st order momentum resolving power for a 1-mm object size (the R52 matrix element). After the inversion of the A matrix to a certain order, it is possible to calculate the final achievable resolution. 2.9. Momentum resolution Considering the set of Eqs. (22) and that the measurements of the observables have uncertainties Dxf ; Dyf ; Dyf ; Dfi ; Dxi ; the final d-resolution of a spectrometer (ignoring straggling) is given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2  2  2  2  2 qd qd qd qd qd Dd ¼ Dxf þ Dyf þ Dyf þ Df þ Dxi ð24Þ qxf qyf qyf qfi i qxi where the derivatives are calculated at a generic point of the phase space. Since d may be expressed as a function of the magnetic rigidity Br Br ¼ Br0 ð1 þ dÞ

ð25Þ

we may write, neglecting the error on Br0 ; DBr Dd ¼ : Br ð1 þ dÞ

ð26Þ

To obtain a magnetic rigidity resolution better than R
1 p ; it is necessary that the following condition is satisfied over all the phase space: 0
1 DdpR
1 p ð1 þ dÞ ¼ Rp

ð27Þ

where we assume a Gaussian distribution of errors in d and the possibility of different charge states are not considered. It is convenient to express the momentum resolution as an effective resolution on the horizontal position Dxeff at the focal plane. This can be done by multiplying each side of Eq. (27) by ðqxf =qdÞ and substituting from Eq. (24), to obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2  2ffi qx qx qx qx f f f f Dxeff ¼ ðDxf Þ2 þ Dxi þ Dyf þ Dyf þ Df qxi qyf qyf qfi i qxf p R0
1 : ð28Þ qd p Because such derivatives can be expressed in terms of the spectrometer aberrations, this result is called aberration-resolution coupling. In the case of well-focused beams with narrow momentum distributions (e.g. Tandem beams), the values of Dxi and R0
1 can be quite small (0.5 mm and 1/5000, say). Thus it is p possible to substitute ðqxf =qxi Þ and ðqxf =qdÞ with their first order values, represented by the horizontal magnification Mx and the momentum dispersion D; respectively. One obtains sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2ffi  2 qxf qxf qxf 2 eff Dx E Dxf þðMx Dxi Þ þ Dyf þ Dyf þ Df qyf qyf qfi i pDR0
1 p :

ð29Þ

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67

Another important source of the broadening of the image is straggling in the target and in the detectors. To obtain a quantitative estimation of the contribution of this to the final image size, one has to consider the coupling between the energetically and spatially broadened beam after passing through material such as detector and target foils, and the dispersion and magnification produced by the magnetic elements. This coupling can be represented as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2  2  2  2  2 qxf qxf qxf qxf qxf Dxj ¼ ð30Þ Dx þ Dy þ Dy þ Df þ Dd qx qy qy qf qd j j j j j where Dxj represents the contribution of the jth layer of material to the image size on the focal plane, and the derivatives are calculated at the position of the jth layer in the spectrometer. When applied to large acceptance spectrometers, the overwhelming term is the chromatic one. In the latter case, a simplified formula can be obtained, as will be discussed later (Section 3.9). For a realistic evaluation of spectrometer resolution one should consider both contributions to the image size, thus obtaining ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2 eff Þ2 ; Dx0 ¼ ðDx Þ þ ðDx j ¼ 1; y; n ð31Þ j j where n indicates the number of different layers traversed by the particles. 2.10. Mass resolution The mass resolving power Rm in a time-of-flight (TOF ) measurement can be estimated from the following non-relativistic expression: qTOF Br ð32Þ M¼ L where M is the mass, q the charge state, TOF the time of flight, L the flight path length and Br the magnetic rigidity of the particles. If particles with different q are separated, for example in a T
xf correlation plot, and hence there is no uncertainty in q; one can write for small deviations of the above values sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2   DM DBr 2 DL DTOF 2
1 ¼ Rm ¼ : ð33Þ þ þ M Br L TOF Taking into account the difference in path lengths for the case of a reaction with a kinematical factor k one obtains ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s      DM DBr 2 ½ðql=qyÞk þ kðql=qdÞ k Dy þ ðql=qdÞk Dd 2 DTOF 2 ¼ : ð34Þ þ þ M Br L TOF Note that the mass resolution depends not only on detector performance (DTOF ) or momentum resolution (DBr), but also on the effect of the aberrations, which could limit it seriously. For ion beams used in nuclear physics research, the effect of the high order chromatic terms is generally small because of the relatively low momentum spread. In that case, only the geometrical aberration needs to be taken into account in the design of a large-acceptance magnetic spectrometer. Note also that such derivatives must be calculated with the appropriate kinematic factor. If the high order terms can be neglected, a simple 1st order formula can be used

DM ¼ M

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s     ffi DBr 2 ðR52 7kR56 ÞDy þ R56 Dd 2 DTOF 2 : þ þ Br L TOF

ð35Þ

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3. Application to the MAGNEX spectrometer This section describes the ion optics calculations for the large-acceptance magnetic spectrometer, called MAGNEX, presently under construction at the INFN Laboratori Nazionali del Sud, Catania. The instrument is intended for the study of nuclear reactions with both stable (primary) and unstable (secondary) beams, the latter coming from an ISOL-type facility EXCYT [2]. The maximum magnetic rigidity has been set principally in view of experiments with beams from the LNS 15 MV Tandem Van de Graff, although some high-energy beams from the K ¼ 800 superconducting cyclotron (SC) may also be accepted. The optical properties of the spectrometer have been investigated by means of numerical raytracing to optimise the magnetic element shapes and field strengths. As concluded in Section 2, in order to compensate the aberrations and the kinematical effect, both hardware and software methods are essential in achieving the required spectrometer performance. The goal of the calculations was to provide a momentum resolving power of about 2000 with a solid angle of about 50 msr and momentum acceptance of 710%. 3.1. General description The spectrometer consists of a vertically focusing quadrupole magnet and a 551 dipole magnet providing the dispersion and the horizontal focusing strength. Details of the technique we used to arrive at the firstorder layout design of the spectrometer are given in ref. [20]. The main parameters resulting from the calculations are shown in Table 1.3 The plan and side projections of the spectrometer, produced by the GEANT simulations [21], are shown in Figs. 4 and 5. The strength for horizontal focusing is achieved by a rotation of both the entrance and exit dipole boundaries b1 and b2 by
181. The value of the momentum dispersion (DB4 cm/% in the focal plane) is chosen to match the foreseen spatial resolution of the focal plane detectors (approx. 0.5 mm) and, on the other hand, to have a reasonable horizontal focal plane size (o100 cm). Assuming a target size of 70.5 mm, with this dispersion and a horizontal magnification Mx of
0.74 (for k ¼ 0), the 1st order momentum resolution from Eq. (4) is estimated to be approximately 5400. However, as is evident from Eq. (23), the real momentum resolution of the device is considerably reduced by the optical aberrations (including those induced by the kinematic factor), by the reconstruction of the position along the focal plane and by the coupling of the dispersion to the energy-straggling in the target or detector material. 3.2. Magnetic elements The main physical and magnetic parameters of the elements are given in Table 2. The maximum magnetic flux density of the dipole is chosen as 1.15 T in order to reduce possible problems of saturation in the pole ends at high field. Two sets of surface coils [14] are inserted between the dipole poles and the vacuum vessel. These surface coils generate quadrupolar (a coil) and sextupolar (b) strength, and are intended to be used for partial compensation of the kinematical effect. The definitions of a and b follow: "    2 # r r BðrÞ ¼ B0 1 þ a
1 þb
1 : ð36Þ r r The entrance and exit boundaries of the dipole pole pieces are profiled by a line with adjoining tangential arcs of circles. Field clamps, shims and surface coils will be used to stabilise and adjust the entrance and exit 3 The linear and angular horizontal magnifications, calculated for k ¼ 0; near the focus and normal to the central trajectory are Mx ¼
0:35 and My ¼
2:95; respectively.

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Table 1 General parameters of the spectrometer. The horizontal magnification, the dispersion and path length specifications are for a kinematic factor k of
0.25 Maximum magnetic rigidity Solid angle Momentum acceptance Horizontal angular acceptance Vertical angular acceptance Deflection angle Focal plane angle Focal plane length Maximum focal plane height Momentum dispersion at the focal plane Horizontal magnification at the focal plane 1st order momentum resolution for 1 mm2 target Path length of the central trajectory

1.8 T m 51 msr 710% (
90, +110) mrad 7130 mrad 551 611 92 cm 32 cm 3.8 cm/% 0.02 C5400 5.76 m

Fig. 4. Plan view of the MAGNEX spectrometer from the GEANT simulations. The trajectories correspond to three different momenta d ¼ 0; 710% coupled with three horizontal angles yi ¼ 0; 770 mrad (fi ¼ 0). The dotted lines enclose the regions of the magnetic fields used in the program.

Effective Field Boundaries (EFB) of the dipole. The pole ends of the quadrupole are designed to be rounded and the fringe field fall-off of the quadrupole will be well clamped to prevent an extension into the target and entrance detector region. The fringe field fall-off at the EFB of both the magnetic elements has been modelled by Enge functions 1=ð1 þ expðP5 ðs=lÞÞ; where s indicates the distance from the EFB, l is the gap parameter and P5 is a 5th order polynomial whose parameters are written in Table 3. The parameters were taken from the measured fringe fields of the bending magnets and the quadrupole of the LNS CS cyclotron extraction line. Other parameterisations with equivalent EFB were tried with little effect on the optics. Extensive field map measurements for all the magnet elements at different excitations will be necessary to effectively implement the method of trajectory reconstruction.

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Fig. 5. Projected side view of the MAGNEX spectrometer. The vertical scale is approximately twice that of the horizontal one. The trajectories were calculated with GEANT and correspond to five vertical angles fi ¼ 0; 765, 7130 mrad coupled with three momenta d ¼ 0; 710% ðyi ¼ 0Þ: The dotted lines enclose the regions of the magnetic fields used in the program.

Table 2 Parameters of the dipole and the quadrupole Dipole Maximum field Bending angle Bending radius, r rmin ; rmax Pole gap Entrance and exit pole face rotation

1.15 T 551 1.60 m 0.95, 2.35 m 18 cm
181

Surface coils Maximum value for a (at 1.15 T) Maximum value for b (at 1.15 T)

0.03 0.03

Quadrupole Maximum field strength Radius of aperture Effective length

5 T m
1 20 cm 58 cm

Table 3 Parameters of the fringe field Enge functions for the dipole and the quadrupole

Dipole Quadrupole

C0

C1

C2

C3

C4

C5

0.503 0.3795

4.43 4.0034


1.39
2.1

0.84 1.1973


0.1590
0.3683

0.0575 0.0478

3.3. Evaluation of aberration-induced effects The effect of aberration terms on physical quantities, i.e. momentum resolution, angle and length of focal plane, etc., was evaluated by use of the ZGOUBI program [22], without consideration of straggling effects.

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Fig. 6. yf 2xf and yf 2xf phase space scatter plots obtained without correction for the aberrations and with k ¼ 0: (ZGOUBI calculations).

This program is implemented with a powerful fit procedure allowing a relatively fast optimisation, which is important, considering the large phase space of the spectrometer. In the initial calculation, a set of particles spanning all the phase space for five different assigned momenta (d ¼
10%,
5%, 0, 5%, 10%), generated by the Monte Carlo routines of ZGOUBI, was tracked through the spectrometer. The beam spot size at the target was assumed to be 1 mm2. In Fig. 6, the yf 2xf and yf 2xf phase space scatter plots without any correction for the aberrations, and for k ¼ 0 are shown. In the yf 2xf plot the lines are very far from the desired vertical, because of the strong contribution of (qxf =qyf ) geometrical aberration. Note also that the shape of these lines is critically dependent on the momentum, indicating a large chromatic component, mainly T126 : Also the scatter plots of the vertical components of the phase space show a non-negligible effect of the vertical chromatic aberration T346 : 3.4. The focal plane angle As discussed previously, the chromatic aberration on the horizontal coordinate can be compensated directly by measurement or indirectly by reconstruction of the position of particles on the inclined focal plane, the angle of which is determined by Eq. (16). As discussed in Section 2.4, unless the detector lies along the focal plane, the reconstruction technique introduces a ballistic error, which we can now quantify for the MAGNEX spectrometer. First, for MAGNEX a focal plane angle of c ¼ 631 is obtained for k ¼
0:25: Furthermore, considering the horizontal angular acceptance and magnification, the yf distribution has a broad range from about
151 to 151. Substituting these values in Eq. (8), one can evaluate the contribution to the image size, for a fixed value of x1 ; as a function of the resolution of the horizontal position and angle. For example, at the edge of the focal plane (x1 ¼ 400 mm) one obtains for the three cases yf ¼ 151, yf ¼ 01 and yf ¼
151 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dxþ E 16ðDx1 Þ2 þ 32ðDyf Þ2 f qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dx0f E 4:4ðDx1 Þ2 þ 2ðDyf Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð37Þ Dxf E 2ðDx1 Þ2 þ 0:5ðDyf Þ2 where the distances are in millimetres, the angles in milliradians and Dx2 is assumed to be equal to Dx1 : Image sizes of several centimetres can be easily result in this way, which would be unacceptable for the

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MAGNEX spectrometer. On the other hand, if the FPD were put along the inclined focal plane, with the same values as above and taking the distance along the particle trajectory between the window and the first position detector s as 70 mm, one obtains from Eq. (9) Dxþ f E0:29Dyf Dx0f E0:14Dyf Dx
f E0:10Dyf

ð38Þ

Eq. (38) shows that the contribution of the straggling of the entrance FPD material to the xf measurement in this configuration are tolerable, being no longer affected by the ballistic error. In Fig. 7 the results of GEANT [21] simulations of the reconstruction of particle position at the focal plane are shown, assuming a 2.5-mm thick Mylar entrance window. The possibility to have the FPD detector at a less extreme angle than that of the focal plane (631) has been explored. The results show how severely the ballistic effect influences the overall momentum resolution of the spectrometer, once the detector is rotated with respect to the focal plane. It appears that a rotation of 51 is the maximum tolerable in order to have a momentum resolving power of around 2000 for the central trajectories. The small k-dependence of the focal plane angle c should be borne in mind, as given by Eq. (16). As k decreases from 0 to –0.5, c decreases from 63.91 to 61.81. Partly for the considerations of the increased window and gas thickness to be traversed by particles at acute incident angles (and it should be remembered that yf extends 151 beyond the central value), but also based on the aberration compensations to be discussed in Section 3.5, it was decided to put the FPD of MAGNEX at 611 as opposed to the 631 given by Eq. (16). In Fig. 8 the result of this compensation is shown. It is clear that second and higher order geometrical aberrations are still present, and these have to be reduced by a careful shaping of the dipole boundaries.

Fig. 7. Illustration of the ballistic effect. GEANT simulations of the reconstructed focal plane image as a function of relative particle momenta (d) and for different initial angles at the target position (shown in separate panels). A 100 mg/cm2 Mylar foil target and entrance start detector were included. The simulated particles were 80 MeV 16O. The symbol key refers to the difference in angle between the detector and the focal plane. The dotted lines are to guide the eye only.

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Fig. 8. The yf 2xf and yf 2xf scatter plots obtained by rotating the focal plane to c ¼ 611: The simulation conditions are the same as in Fig. 6.

3.5. Corrections with entrance and exit dipole boundaries The strong effect of geometrical aberrations coupled to the large phase space made the search for the best profile of the dipole rather difficult. Furthermore the design must allow tolerable values for ymax and the f vertical chromatic aberration. Initially, the possibility to compensate such effects by a parabolic profiling of the bending magnet pole surfaces was explored. The unreasonably high values of the field indexes that were necessary, producing themselves high-order aberrations, made this choice unfeasible. However, it was found that the shape of the EFB strongly influenced the focal properties of the spectrometer. Traditional techniques to shape EFBs by polynomial functions with hand-adjusted parameters appeared rather inconvenient, due to the large phase space to be compensated. The use of automatic procedures for the optimisation based on minimisation algorithms, available in the ZGOUBI program [22], circumvents this laborious task. In particular, the EFB profiles are described as splines composed of straight lines and tangential circular arcs with a total of four free parameters for each boundary. An often-used technique for the fit procedure is to choose the values of certain high order matrix elements (aberrations) as the parameters of the fit. This technique is more reliable if there are few aberrations known to be dominant, which is typical for small acceptance devices. A set of calculations based on this demonstrated that for MAGNEX, even if convergence is achieved, a poor momentum resolution is obtained because of uncompensated aberrations. For these reasons a special technique, suited for application to large acceptance spectrometers, was used. It is still based on the automatic optimisation procedure of ZGOUBI but now the parameters of the fit are the required position and angles of particles at the focal plane. The advantage of this technique is that the overall effect of aberrations is intrinsically minimized, no matter how they individually contribute to the deterioration of the image. Also the ray-tracing and fit procedure is acceptably fast and simple to change. After the parameters of the EFBs are so defined, they are input to a Monte Carlo calculation in order to view the full phase space with good statistics. In Fig. 9 we show the results. The strong improvement in the focus is accompanied by a significant reduction of both ymax and the f focal plane vertical size. 3.6. Correction for kinematic line broadening As discussed in Section 2.6, the simplest first order compensation of the kinematic effect is to shift the focal plane detector by an amount given by Eq. (18). The left panel of Fig. 10 shows the change in momentum dispersion DDx ; horizontal magnification DMx and the shift of the horizontal focus DL as a function of k: For direct nuclear reactions, typical values of jk| range between 0 and 0.5, as shown in the

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Fig. 9. ðxf ; yf Þ scatter plot at k ¼
0:25 after optimisation of the EFBs. Simulation conditions are as in Fig. 6.

Fig. 10. Left panel: The horizontal magnification Mx ; the momentum dispersion variation DDx and the focal length shift DL versus the kinematic coefficient k: Right panel: Kinematic factors jkj calculated for various reactions plotted against centre-of-mass angle ycm : The reactions are: 4He+39K-4He+39K* (Einc ¼ 45 MeV, Ex ¼ 3 MeV), 12C+12C -12C + 12C (Einc ¼ 60 MeV) and 11Be + 1H - 10Be* + 2H (Einc ¼ 388 MeV, Ex ¼ 3:37 MeV). The latter is an actual radioactive beam experiment [23] performed with the SPEG spectrometer at GANIL.

right panel of the same figure. Values in excess of unity could in principle be obtained for inverse kinematic reactions, but a practical limit is set by the fall-off of cross-section at large centre-of-mass angles. As seen in the left panel of the figure, the linear magnification Mx becomes unacceptably large for increasing positive values of k: High-order calculations with ZGOUBI confirm this trend, giving at k ¼ 0:25 a value of Mx E
1:7: Applying the latter to a beam spot size of 1 mm, one obtains a contribution to the image size of about 1.7 mm. On the contrary, much better optical properties are found for negative k values. An important aspect of 2nd and higher order kinematic compensation is that the EFB are fixed while the value of k depends on the particular reaction. Thus one can only compensate for one value of k with a fixed EFB. Otherwise one could calculate EFB shapes for a range of different k values in order to construct dismountable pole tips to use for different experiments. Considering all of the above, it was decided (i) to have the spectrometer bend to the right with spectroscopic measurements usually taking place on the left-side of the beam, and (ii) to optimise the shape of the dipole EFB and of the distance between dipole and focal plane detector to compensate ‘‘by default’’ a k of
0.25. A range of k of 70.25 centred at
0.25 is to be compensated by adjustable means. In Fig. 9 we show the results of optimisation for k ¼
0:25; obtained by a first order shift of the focal plane by
16.4 cm from the position of at k ¼ 0; together with a higher order compensation by re-shaping the dipole EFB. From that default configuration of the spectrometer, the first order compensation for different values of k is

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75

obtained by simultaneous action of the a surface coil and an additional shift of the focal plane detector. Without the a coil, the value of DL in the fixed interval would be 70.15 m, as may be read from the graph in the left panel of Fig. 10. The use of the a coil is thus to reduce this inconveniently large shift at maximum magnetic rigidity and to compensate it completely for k between –0.2 and –0.3, as shown in Fig. 11. For small acceptance spectrometers, the first order correction of k is often adequate. However, as one can see from Fig. 12, even at the new detector position the foci for MAGNEX are strongly affected by the 2nd and higher order induced aberrations, as indeed expected from Eq. (15). The term 2k T126 gives the main contribution to the 2nd order effect. From Eq. (14), one obtains the following estimation for the derivative (qxf =qyf ):   qxf T126 þ kT166 : ð39Þ E2kyi qyf R22 This equation allows estimates of (qxf =qyf ) over a broad range of y; once the appropriate aberrations are calculated.

Fig. 11. Focal plane shift as a function of the kinematic coefficient k in the range covered by the MAGNEX spectrometer. The three lines refer to different values of the applied field index produced by the a coil. The squares are the shift with a ¼ 0:03 which is the maximum value possible at maximum main field strength (1.15 T). The triangles and the circles correspond to the maximum a strength at 0.6 and 0.2 T, respectively.

Fig. 12. (xf ; yf ) scatter plot at k ¼
0:5 without kinematic correction (left panel) and after the 1st order correction by an a-coil excitation and focal plane shift (right panel). Simulation conditions are as in Fig. 6.

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3.7. Software corrections Simulations of the software corrections were performed with the program COSY INFINITY by the use of the reconstruction algorithm described in Section 2.8. The simulation loop included [24] 1. 2. 3. 4.

generation of initial coordinates of particles uniformly distributed within the acceptance of the device, tracking through the spectrometer field map connecting initial and final phase space coordinates, addition of uniformly distributed measurement errors to the angles and coordinates in the focal plane, application of the 1st–5th order reconstruction algorithms to determine the momenta of particles.

The momentum resolution was calculated at the end of the simulation loop in a direct way, from the differences between simulated and reconstructed momenta. We emphasise here the importance of using the initial vertical angle fi rather than ff in the focal plane. This arises from the small value of the vertical angular magnification Mf ¼ ðqff =qfÞB1=9 for d ¼ 0 which results to first order in a nine times larger value of the derivative (qxf =qff) compared with (qxf =qfi ) !     qxf qxf qxf
1 Mf E9 : ð40Þ ¼1 qff qfi qfi As a result, to keep to the same final momentum resolution, the error in the measurement of ff must be one-ninth that of fi ; i.e. o1 mrad, which is hardly feasible in view of detector straggling effects. One should add that the vertical angular magnification is perturbed by higher order chromatic aberrations (see Fig. 13) such that for extreme momenta jdj ¼ 10% it reaches the order of two.

Fig. 13. (ff ; fi ) scatter plot obtained by raytracing at three fixed momenta:
10% (the upper-left to bottom-right line), 0% (the central line) and +10% (the bottom-left to upper-right line) demonstrating a momentum-dependent correlation between the initial fi and the final ff vertical angles. xi ¼ yi ¼ 70:5 mm,
90 oyi o110 mrad,
130ofi o130 mrad, all uniformly distributed.

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Fig. 14. Bar chart showing the momentum resolving power of the spectrometer after the reconstructive correction for aberrations versus the reconstruction order. The results for Dyf ¼ 72:0 mrad (open bars) and Dyf ¼ 76:0 mrad (shaded bars) and from right to left Dxf ¼ 70:3; 70:6; 71.0 mm.

Fig. 14 shows the achievable resolutions of the spectrometer for different reconstruction orders and for a variety of possible horizontal angle and position measurement errors. The other measurement errors in the calculations were taken as Dyf ¼ 70:5 mm and Dfi ¼ 73:5 mrad. The initial coordinates of particles were distributed uniformly all over the total acceptance of the spectrometer: xi ¼ yi ¼ 70:5 mm,
90oyf o110 mrad,
130ofi o130 mrad, d ¼ 710%: To simulate the measurement of the initial vertical angle fi instead of a final ff ; the error of the ff measurement was set to Dff ¼ Mf Dfi. That is a 1st order approximation and overestimates Dff for non-central momenta. Thus a direct measurement of fi should give better results than those presented in Fig. 14. 3.8. Estimation of straggling-dispersion effect As discussed in Section 2.9, a potentially significant source of image broadening comes from straggling and small angle scattering in material along the particle path. In Section 3.4 the effect of the entrance window of focal plane detector has been quantified, but not that of the target and other material situated before the dipole. For the latter, it is necessary to consider the global effect produced by the coupling of energy-straggling with the dispersion. On the basis of Eq. (30), we will estimate the effects of a 100 mg/cm2 Mylar target (0.72 mm) and that of a 0.8 mm 12C equivalent emissive foil foreseen for the MAGNEX microchannel-plate start detector (PSD) [25,26]. The straggling and small angle scattering distributions have been calculated with SRIM-2000 [27] for two beam types and energies, typical of those for which MAGNEX is intended, while the derivatives involved in Eq. (30) were obtained either from a fit to phase space scatter plots or, in the case of the PSD foil, by a linear scaling of the 1st order transport matrix elements. More details may be found in Ref. [28]. The results are given in the first part of Table 4. The overwhelming contribution comes from the momentum-dispersive term, the other contributions being at least two orders of magnitude smaller. Taking the entrance PSD as an example, a good approximation is therefore   qxf DxPSD E DdED Dd ð41Þ f qd PSD where D is the full dispersion4 on the inclined focal plane, because the PSD is before any dispersive element. Similar considerations are valid for the target. We have neglected the higher order chromatic terms in the 4

Dispersion to which an object put in the target position is subjected.

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Table 4 Evaluation of the contribution to the final momentum resolution from the entrance PSD detector and a 100 mg/cm2 Mylar targeta Ion E (MeV) Dd (%) Dx (mm) Dy (mm) Dy (mrad) Df (mrad) Global PSD effect (mm) GEANT (PSD only) (mm) Global target effect (mm) GEANT (target only) (mm)

7 Li 49 0.027 o10
5 o10
5 o10
2 o10
2 0.9 0.9 0.7 0.5

16 O 135 0.019 o10
5 o10
5 0.2 0.2 0.8 0.9 0.7 0.8

a The first five rows of values show the size of the contributing terms to Eq. (29) calculated by SRIM-2000. In the lower part of the table, the effect of the PSD and target at the focal plane (‘‘Global effect’’) calculated by Eq. (29) with a momentum dispersion of 38 mm/% are given, together with the results of GEANT simulations.

final approximation of Eq. (41). As a cross-check of our approximations, we may compare to full simulations with the same target and entrance PSD by the GEANT program,5 which are given in the last part of Table 4. From these calculations we conclude that if we need a global contribution to the final image size of o2 mm, the energy spreading in the target and start detector should not exceed 0.09%. Considering that a typical real target foil would hardly be thinner than the one assumed here, the energy-straggling limits the PSD start detector to a total thickness (active and dead layers) of no more than 0.8 mm equivalent carbon. This precludes the use of gas detectors, for example, as the PSD start system, because of the necessary window and gas thickness. 3.9. Momentum resolution and angle measurement errors In order to estimate the required precision on the angle measurements that will allow a momentum resolution of one part in 2000, we refer to Eq. (28). However, we must also include the effect of straggling in the target and entrance PSD, for which we assume a contribution to the image size of 1.5 mm (estimated from combining the results given in Table 4) and add in quadrature to the other factors, following Eq. (31). We take values of D and Mx from table 1, noting that the latter is especially sensitive to k: For the moment, we only consider our ‘‘central’’ k value of
0:25 ¼ k0 : We further assume a beam spot size xi of 1 mm, and a focal plane detector intrinsic resolution xf of 0.5 mm. We concentrate on the d ¼ þ5% region of the focal plane, where we particularly want to achieve a good resolution, namely R0p ¼ 2100: With these values, the following condition must be satisfied: s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2  2 qxf qxf qxf Dyf þ Dfi þ Dyf p0:88 mm: ð42Þ qyf qfi qyf The values of the derivatives were obtained by examining scatterplots of the relevant variable against xf ; as produced by the ZGOUBI program with d ¼ 5% and fi ¼ 0 mrad. The maximum absolute values of the

5 Note, however, that GEANT uses empirical energy-loss and straggling formulae deduced from systematics, which may explain the differences with the SRIM-2000 case-by-case results.

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derivatives ðqxf =qyf Þ; ðqxf =qfi Þ and ðqxf =qyf Þ are estimated as    qxf     qy  E0:032 mm=mr f max    qxf     qf  E0:044 mm=mr i max    qxf     qy  E0:24: f max

ð43Þ

If we assume an accuracy in Dyf of 1 mm, one can obtain from condition (42) the ‘‘allowed’’ errors of measuring yf and fi in the specified d ¼ 5% region for k ¼
0:25 are given by 1:0210
3 Dy2f þ 1:9410
3 Df2i p0:72 mm2

ð44Þ

where the errors are FWHM values (in mrad) of the distributions. This allows rather generous errors for yf and fi (about 15 mrad for each). A position sensitive detector located 200 mm from a 70.5 mm target and having a spatial resolution of 1 mm, would result in Dfi E7 mrad. The error on the yf measurement mainly depends on the angular straggling in the entrance window of the focal plane detector. Estimations of the straggling in the FPD, performed with SRIM-2000 [27], yield values of Dyf generally within 75 mrad. These measurement errors amply satisfy the condition (44). Now we turn to values of k different to the central value for which the spectrometer has been optimised. Firstly, as one approaches either of the limits k ¼ 0 or k ¼
0:5; the magnification Mx becomes large (see Fig. 10), and the Mx Dxi term in Eq. (29) begins to contribute significantly to the resolution. Secondly, there are important k-induced aberrations that contribute to ðqxf =qyf Þ: This derivative may be estimated according to Eq. (39), rather than the one given in Eq. (43) above for k0 : One obtains for the MAGNEX spectrometer   qxf ð45Þ E0:7yi ðk
k0 Þ mm=mr: qyf In the numerical estimation of Eq. (45), the small term proportional to k2 is neglected. This gives a rough value for the range of k within which the 2nd order kinematic effect can be software-compensated and still achieve the desired resolution. If we take the empirical angle errors estimated above, Dyf E10 mr and Dfi E7 mr

ð46Þ

we find that, the range of k beyond k0 cannot exceed 0.05. This would be very restrictive, and one must turn to hardware corrections. To partially compensate for Eq. (45), a b correcting coil is placed on the pole surfaces of the dipole to create a 2nd order component in the radial field expansion. The ray-tracing results with such a compensation for k ¼
0:5 are shown in Fig. 15 (again, one should concentrate on d ¼ 5%). One observes a chromatic dependence of this compensation, especially at maximum yf : This probably arises because the coils extend through the dipole until its dispersive exit. As a result, some small region (about 10% on the negative yi side, corresponding to yf > 1250 mrad), where the correction is not satisfactory, remains unavailable for trajectory reconstruction with the required accuracy. In most of the region, however, the limiting yf and fi measurement errors are close to the ones estimated to be achievable. For
0:35oko
0:05; a momentum resolution of 1/2000 is obtainable over the full angular acceptance, with the b coil suitably excited.

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Fig. 15. ðxf ; yf Þ scatter plot after the 1st and the 2nd order correction with the a and b surface correcting coils of the dipole. Simulation conditions are similar to those in Fig. 6, but with k ¼
0:5:

3.10. Mass resolution As discussed in Section 2.9, the 1st order mass resolving power Rm in the TOF measurements can be estimated in the case of kinematical effects from the simple formula (35). The first term of that equation, concerning (DBr=Br) is o10
3 and may be neglected. Substituting the values of the matrix elements for the MAGNEX spectrometer and considering a k value of
0.25 one obtains for the 2nd term (DL=L) of Eq. (35) DL E10
3 ð0:91 Dy þ 0:37 DdÞ L

ð47Þ

where Dy is in mrad and Dd is a %, as usual. The chromatic contribution can be neglected because of the small value of initial momentum spread (Ddo0:2% for beams from most accelerators). By contrast, the angular dependence of DL=L is strong. Assuming for the entrance PSD a spatial resolution of 1 mm at a distance of 200 mm from a 70.5 mm target, an accuracy in the yi measurement of DyB7 mrad is obtained, which gives DL=LB1=130: However, assuming Dyf B10 mrad measured at the focal plane, the reconstructed initial angle resolution would be Dyi B4 mrad, giving a much more favourable DL=LB1=230: For the calculation of the DT=T term of Eq. (34) the simple formula DTOF ¼ TOF ðA þ 1Þ
TOF ðAÞ ¼

1 TOF A

ð48Þ

was used, where TOF ðAÞ and TOF ðA þ 1Þ indicate the time of flight of two ions having the same kinetic energy and atomic number A and A þ 1; respectively. To be resolved, the DTOF must be > ADt; where Dt is the time resolution of the detection system. As an example, for the case of 40Ca at 140 MeV TOF is about

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81

Fig. 16. TOF separation between mass 38 and mass 39 Ca isotopes, as simulated by GEANT for the MAGNEX spectrometer. A yf cut of 10 mrad has been applied.

230 and the above condition is well satisfied for Dt ¼ 1 ns. Such a time resolution is obtained by the proposed system of MCP start detector (B0.5 ns) and Silicon wall detector (B0.8 ns) for the MAGNEX project [25]. Thus, errors in the L and TOF measurements give the main contribution to the mass resolution DM=M: For DTOF =TOF B1=200 and DL=LB1=230; one obtains DM=MB1=150: Even better values may be obtained for slow heavy ions, e.g. DM=MB1=180 for 40Ca at 80 MeV with Dt ¼ 1 ns. As a test of the above estimates, simulations were performed with GEANT. A sample of 38Ca18+ and 39 Ca18+ ions were generated before the target with energies ranging between 140 and 100 MeV and sent round the spectrometer. The time difference between the entrance PSD and the stopping silicon array in the FPD, with individual time resolutions as given above, is stored in a histogram (see Fig. 16). A Dyf cut of 10 mrad has been applied in order to reduce the y dependence of the TOF : One observes a mass resolution DM=M of B1/130, in agreement with the estimations above.

4. Discussion and summary It is pertinent to compare with previous work on large solid angle spectrometers. In the past, magnetic spectrometer design for low-energy heavy ions, such as given by Tandem beams, has tended to emphasise high momentum resolution rather than large solid angle O and momentum acceptances. Examples are the Q3D series [10] (O ¼ 15 msr), the Oxford MDM2 [13], the ENMA spectrometer for the JAERI tandem [12,29] and the QDD spectrometer at the TANDAR laboratory [30]. As discussed in Section 3.8, the effect of the target significantly contributes to the achievable resolution at low beam energy. Thus the experimentally observed momentum resolutions for such spectrometers are typically 5  10
4. The Q3DII has a large momentum dispersion (10.2 cm/%), allowing an aberration-limited resolving power as large as 10 000, but the consequence of such large dispersion is that the focal plane length at full momentum acceptance (710%) is about 2 m and is difficult to cover with a practical high-resolution detector.

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At intermediate beam energies, the effects of straggling in the target and detector foils are less significant and the recent trend has been to design spectrometers that rely on ray-reconstruction techniques to achieve the momentum resolution goals. Examples of these are the O = 20 msr ‘‘Large Acceptance Spectrometer’’ [31,32] at RCNP, Osaka (used as a light-particle counterpart arm to the high-resolution spectrometer ‘‘Grand Raiden’’ [33]) and the superconducting vertical spectrometer ‘‘S800’’ at NSCL, MSU [11,34]. Although other magnetic spectrometers have been designed and built with solid angle acceptances as large or even larger than MAGNEX, they have been intended primarily for mass spectroscopy and not for energy or momentum resolution per se. One of the first proposed magnetic spectrometers with an acceptance >50 msr was SUSAN [35] which was designed as a superconducting solenoid followed by a single dipole. It was intended as a mass analyser for heavy ions (A ¼ 802200) emitted in two-body collisions. The estimated momentum resolution was somewhat better than 5  10
3, which, following Eq. (33), is sufficient to give the required mass resolution of 1 in 200. A derivative of SUSAN is VAMOS (VAriable MOde Spectrometer) [4] which is near to completion at GANIL. One of the modes of VAMOS is momentum-dispersive (DE2:5 cm/%), again, using software to reconstruct the image. The momentum acceptance of VAMOS is about one half that of MAGNEX. On the other hand, the solid angle can be increased up to a maximum of 100 msr by putting the target as close as 40 cm to the entrance of the first quadrupole. In summary, the optical properties of a large acceptance, high-resolution magnetic spectrometer have been studied extensively. The strong second and higher order aberrations were satisfactorily compensated by an appropriate shaping of the entrance and exit EFB of the dipole using a new technique for reconstructing the field shape parameters. A software reconstruction algorithm was used to simulate the final achievable momentum resolution. The constraints on the resolutions in the measurements of the positions and angles were determined. A final result of about one in 2000 for the momentum resolution is obtained with a 5th order reconstruction of the trajectories. The corrections were extended to a broad range of kinematic coefficients (
0:5oko0) by shifting the focal plane detector and the introduction of surface coils between the dipole poles. For tandem beams, the mass resolution is demonstrated to be 1/150 or better, depending on the particle energy.

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