Collimation and decoupling of ECR source beamsfor brilliance optimization

C. Xiao a,n, L. Groening a, O.K. Kester a,b

a GSI Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germanyb Institut für Angewandte Physik, Goethe Universität, D-60483 Frankfurt am Main, Germany

a r t i c l e i n f o

Article history:Received 5 September 2013Received in revised form22 November 2013Accepted 23 November 2013Available online 3 December 2013

Keywords:ECR sourceCollimationDecoupling

a b s t r a c t

The four-dimensional transport of the transverse phase space of the extracted beam was calculated forthe CAPRICE electron cyclotron resonance (ECR) ion source at GSI. It is especially of interest for an ECRion source, where asymmetric beams of the transverse phase spaces are extracted in the presence of astrong solenoidal field, and this axial magnetic field adds an angular momentum to extracted beam,resulting in a strong horizontal and vertical (x–y) coupled beam. The paper presents multi-particletracking simulations using the well-established particle-in-cell code BEAMPATH. The results illustratethat the beam brilliance can be improved by combination of multi-stage collimation with skewquadrupole decoupling.

Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved.

1. Introduction

Ion beams extracted from ECR ion sources have a complicatedstructure in four-dimensional phase space [1,2]. Prior to extractionthe beam inside the plasma chamber is round (ɛx ¼ ɛy) anduncoupled. A particle distribution with equal projected rmsemittances but with non-equal eigen-emittances in phase spaceis strongly coupled after extraction, the beam is produced [3].

A collimation channel which can be adjusted to allow thetransport of beams with a certain beam emittance, is an ideal toolto optimize the ECR ion source tuning in terms of beam brightness. Asolenoid collimation channel comprising three solenoids and fourapertures has been built at the SuSI injection line at MSU [4], firstexperimental studies of the collimation principle have been com-pleted successfully. Since a collimation channel is a powerful tool forion source tuning, GSI is aiming for this principle with respect to thefacility for antiproton and ion research (FAIR) facility [5]. In our case,a quadrupole collimation channel is selected to limit the beamemittance in transverse phase space in order to avoid beam loss inthe following machines. Downstream of the quadrupole collimationchannel, the decoupling section composed of a skew quadrupoletriplet is applied to remove the inter-plane correlations [3,6].Projected rms emittances become equal to the eigen-emittances,which means that they become different from each other.

The paper starts with an introduction of the relationship betweenthe projected rms emittance and eigen-emittance. Afterwards the

beam transport setup for multi-stage collimation and transverseemittance transfer is presented. In the fourth section, simulations forthe charge-to-mass selection system are presented. Afterwards, theprinciples and effects of the quadrupole collimation channel, theeigen-Twiss parameters, and the transverse acceptance of this channelare investigated. The sixth section is on comparison between front-to-end simulations of the low energy beam transport with and withoutcollimation. Afterwards, the reconstruction of the phase space byinter-plane correlation parameter measurements is illustrated. The lastsection draws some conclusions and gives an outlook.

2. Projected and eigen-emittances

The full four-dimensional symmetric beam's second momentsmatrix contains ten unique elements, four of which describe thehorizontal and vertical coupling. If at least one of the elements ofthe off-diagonal sub-matrix is non-zero, the beam is x–y coupled:

C ¼

⟨xx⟩ ⟨xx′⟩ ⟨xy⟩ ⟨xy′⟩⟨x′x⟩ ⟨x′x′⟩ ⟨x′y⟩ ⟨x′y′⟩⟨yx⟩ ⟨yx′⟩ ⟨yy⟩ ⟨yy′⟩⟨y′x⟩ ⟨y′x′⟩ ⟨y′y⟩ ⟨y′y′⟩

266664

377775: ð1Þ

In here, we use x and y to indicate transverse particle displace-ments and x′ and y′ to indicate their derivatives with respect to theposition s along the design orbit. Projected rms emittances ɛx andɛy are quantities which are used to characterize the transversebeam quality in the laboratory coordinate system and are invariantunder linear uncoupled (with respect to the laboratory coordinatesystem) symplectic transformations. Projected emittances are the

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Nuclear Instruments and Methods inPhysics Research A

0168-9002/$ - see front matter Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nima.2013.11.084

n Corresponding author. Tel.: þ49 6159711961.E-mail address: c.xiao@gsi.de (C. Xiao).

Nuclear Instruments and Methods in Physics Research A 738 (2014) 167–176

rms phase space areas covered by the projections of the particlebeam onto each plane, and their values are equal to the squareroots of the determinants of the on-diagonal sub-matrices.

Eigen-emittances ɛ1 and ɛ2 are quantities which give beamdimensions in the coordinate frame in which the beam matrix isuncoupled among degrees of freedom and are invariant underarbitrary (possibly coupled) linear symplectic transformations.None of the projected emittances can be smaller than the smallesteigen-emittance. Diagonalization of four-dimensional symmetricbeam's second moments matrix yields the eigen-emittances, andtheir values can be expressed as follows [6]:

ɛ1 ¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�tr½ðCJÞ2�þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitr2½ðCJÞ2��16 detðCÞ

qrð2Þ

ɛ2 ¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�tr½ðCJÞ2��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitr2½ðCJÞ2��16 detðCÞ

qr: ð3Þ

The four-dimensional beam matrix J is the skew-symmetricmatrix with non-zero entries on the block diagonal off form:

J ¼

0 1 0 0�1 0 0 00 0 0 10 0 �1 0

26664

37775: ð4Þ

If the beam matrix is already uncoupled in the laboratory frame,the set of projected beam rms emittances coincides with the set ofeigen-emittances. If the beam matrix has inter-plane correlationbetween horizontal and vertical phase planes, these two sets aredifferent. This feature has interesting applications in acceleratorphysics and gives the theoretical basis for the round-to-flat transfor-mations of angular momentum dominated beams [7–10].

The beam brilliance, a figure of merit for a beam quality usefulfor high-current low-emittances beams was originally introducedas follows:

Bn ¼ IV4

ð5Þ

where I is the beam current and V4 is the hyper-volume in the fourdimensional transverse phase space occupied by the beam parti-cles [11]. This definition is mathematically straight forward but oflimited use for an accelerator-designer: the four-dimensionalemittance might be very small but if the projected emittancesare enlarged due to coupling, these projected emittances exceedthe horizontal and/or vertical acceptance of the channel and thebeam is partly lost. Accordingly, for practical use, we present arefined definition as follows:

B¼ Iɛxɛy

: ð6Þ

The beam brilliance is expressed in terms of the beam currentand the projected rms emittances of their respective distributions.Unlike original brilliance Bn, the redefined brilliance B can beincreased by decoupling using skew quadrupoles and/or solenoids.

3. Beam transport setup

Fig. 1 displays the layout of the beam transport setup, and thetotal beam line can be divided into the four sections. At the end ofthe beam line, a scintillator screen is placed to measure x–y beamprofiles [12,13], and an Allison type emittance scanner is added tomeasure two-dimensional x–x′, y–y′ phase spaces [14]. The totallength of beam line is 12 m. The sections are:

I: Charge-to-mass selection section, from the ion source exit tothe horizontal slit (charge resolving plane). It comprises a solenoidand a horizontal bending magnet.

II: Matching section, from the horizontal slit to the first collimator.It comprises two normal magnetic quadrupole doublets.

III: Quadrupole collimation channel, from the first collimator tothe last collimator. It comprises three normal magnetic quadru-pole doublets.

IV: Decoupling section, from the last collimator to the end ofthe beam line. It comprises two normal magnetic quadrupoledoublets, one skew magnetic quadrupole triplet, and another twonormal magnetic quadrupole doublets. If the skew magneticquadrupole triplet is turned on, this section is adopted to decouplethe beam. The gradients of the first two magnetic quadrupoledoublets and skew magnetic quadrupole triplet are optimized toremove the inter-plane correlation. If the skew magnetic quadru-pole triplet is turned off, this section is adopted to rematchthe beam.

Extraction from the ECR ion source and transport through thebeam line are simulated with the particle-in-cell code BEAMPATH[15]. In simulations, the solenoid magnetic field is the interpola-tion of two-dimensional grid functions Bzðzi; rjÞ, Brðzi; rjÞ and thefield map is calculated by the program OPERA-2D [16]. Themagnetic field inside the horizontal bending magnet is describedby the Taylor expansion up to the second order terms [17]:

Bxðx; y; zÞ ¼ By �nyRþ2ζ

xy

R2

� �� �ð7Þ

Byðx; y; zÞ ¼ By 1�nxRþn2y2

R2þζx2�y2

R2

� �� �ð8Þ

where By is the vertical component of magnetic field along thereference trajectory with curvature radius R, n is the field index,and ζ is a non-linear coefficient in the magnetic field expansion:

n¼ � RBy

∂By

∂xjx ¼ 0;y ¼ 0 ζ ¼ R2

2By

∂2By

∂x2jx ¼ 0;y ¼ 0: ð9Þ

At the entrance and exit of the analyzing magnet, the slopes ofthe particle trajectories are changed because of the pole angle αaccording to the linear matrix transformation [17]:

x

x′y

y′

266664

377775¼

1 0 0 0tgαR 1 0 00 0 1 00 0 � tgðα�ϕÞ

R 1

266664

377775

x0x0′y0y0′

266664

377775 ð10Þ

where the correction angle ϕis given by the expression [20]:

ϕ¼ k1gR

� �1þ sin 2αcos α

1�k1k2gR

� �tgα

h ið11Þ

where g is the gap of the analyzing magnet and the coefficients k1,k2 are defined by the pole geometry. Typical values of k1 and k2 fora quare-edged magnet are 0.45 and 2.8, respectively.

The magnetic field of a multi-pole lens of the order n (n¼2 formagnetic quadrupoles) is represented as a step function in thez-direction with field components:

Br ¼ �B0rR0

� �n�1

cos ðnθþθ0Þ ð12Þ

Bθ ¼ B0rR0

� �n�1

sin ðnθþθ0Þ ð13Þ

where B0 is pole tip field, R0 is a pole radius, and θ0 is a skew angle(θ0¼01 for normal magnetic quadrupole, and θ0¼�451 for skewmagnetic quadrupole).

C. Xiao et al. / Nuclear Instruments and Methods in Physics Research A 738 (2014) 167–176168

4. Charge-to-mass selection

Fig. 2 shows the calculated particle distribution for O3þ16

extracted from the CAPRICE ECR ion source (see Fig. 3) at anenergy of 2.5 keV/μ. In order to minimize beam losses andaberrations, a short solenoid is used directly after the extraction

system of the ECR ion source to provide simultaneous focusing inboth planes. A double-focusing analyzing magnet behind thesolenoid is employed to separate the required charge state. Inour previous work [3] the simple solenoid matrix formalism isselected for simulation. We treated cut A as ECR ion source exitand the particle distribution of the different projects is calculated

Fig. 1. CAPRICE beam transport setup.

Fig. 2. Four-dimensional phase space distribution at the exit of CAPRICE ECR ion source (cut B of Fig. 3). Horizontal and vertical rms Twiss parameters are indicated as well asthe projected rms emittances.

C. Xiao et al. / Nuclear Instruments and Methods in Physics Research A 738 (2014) 167–176 169

systematically [2]. In the present simulations the two-dimensionalsolenoid magnetic field map is adopted for simulation and thefringe field length is considered. Therefore, ECR ion source exitstarts from cut B (approximately 15 cm inside of the ion source).Since the magnetic field along the axis of the ECR ion source isrelatively weak (comparing with the peak solenoid field) betweencut A and B, we treat this beam extraction region as a drift spaceand perform backward particle tracking from cut A to B.

In many cases there is no correlation between the four trans-verse coordinates (x, x′, y, y′) and the longitudinal coordinates (l, δ),so that the phase space density can be factorized into a transverseand longitudinal one: ρ(x, x′, y, y′, l, δ)¼ρ(x, x′, y, y′)ρ(l, δ). For manybeam line elements, e.g. drift spaces, einzel lenses, quadrupoles, andbending magnets, the four-dimensional transverse phase spacedistribution ρ(x, x′, y, y′) can be separated in horizontal and verticaltwo-dimensional distributions: ρ(x, x′, y, y′)¼ρ(x, x′)ρ(y, y′). How-ever, solenoids and skew quadrupoles couple the two transversephase spaces and for these elements one has to consider the fulltransverse phase space distribution ρ(x, x′, y, y′). As in this paperwe only consider beams with very small momentum spread(δp=po10�4). Therefore, no dispersion compensation is appliedwith the analyzing magnet.

The four-dimensional phase space distribution in Fig. 2 is con-siderably correlated between the horizontal and vertical phaseplanes. This distribution is plotted through six two-dimensionalprojected distributions, e.g. the horizontal and vertical emittancedistributions ρðx; x′Þ and ρðy; y′Þ, spatial distribution ρðx; yÞ, crosscoupled emittance distributions ρðx; y′Þ and ρðx′; yÞ, and angularcorrelated distribution ρðx′; y′Þ. Since the total beam current is some10 eμA, space-charge forces are negligible. In a rough approximation,we assume that the initial four-dimensional particle distribution issimilar for O2þ

16 , O3þ16 , and O4þ

16 ions, and the solenoid focusingstrength and beam rigidity inside of bending magnet are set forO3þ16 . The purpose of the solenoid is to adjust the angle of the beam

going into the analyzing magnet. Since the actual beam diametercannot be controlled with a single solenoid, a sufficiently largemagnet gap (g¼9 cm) has to be chosen. Transport of multiple chargestate beams is calculated to verify the analyzing ability of thebending magnet. The horizontal beam trajectories from the ECRion source exit to the charge resolving plane are illustrated in Fig. 4.Separation of different charge states at the charge resolving plane isdemonstrated, showing sufficient separation of the charge states.

Separation also works fine quite independent of the distributions ofO2þ16 and O4þ

16 medium mass ions.

5. Quadrupole collimation channel

The quadrupole collimation channel is placed behind theanalyzing magnet in order to improve the beam brightness. Anuncorrelated beam with Twiss parameters (αx¼�1.15, αy¼1.15,βx ¼ 1:18 m=rad and βy ¼ 1:18 m=rad) is assumed at the entranceof the quadrupole collimation channel being the periodic matchedsolution of the channel. By setting calculated values for thegradients (39 T/m for O3þ

16 ) as well as for slit apertures(4 cm�4 cm), only particles in a defined phase space volumeare transmitted through the entire channel, all other particles arestopped at the apertures along the channel. Multi-particle trajec-tories with and without collimators through the quadrupolecollimation channel section are shown in Fig. 5.

Fig. 3. Cross-section of the CAPRICE ECR ion source. The ion beams are extractedfrom right to left. Beam extraction is through the exit fringe of its solenoid. This exitfringe changes the eigen-emittances and creates inter-plane correlations.

Fig. 4. Horizontal multi-particle trajectories for O2þ16 , O3þ

16 , and O4þ16 ions from an

ECR ion source exit (z¼0 cm) to the charge resolving plane (z¼180 cm). Differentcharge states have the same initial transverse coordinates (x, x′, y, y′) but thedifferent initial beam dimensionless z-momentums (βγ) at the exit of ECR ionsource.

Fig. 5. Multi-particle trajectories through the quadrupole collimation channel, fourconstant collimators are located at z¼0, 90, 180 and 270 cm. Blue lines are thebeam trajectories without collimation, and red lines are the beam trajectories withcollimation. (For interpretation of the references to color in this figure caption, thereader is referred to the web version of this paper.)

C. Xiao et al. / Nuclear Instruments and Methods in Physics Research A 738 (2014) 167–176170

Fig. 6. The particle distributions in the x–x′ and y–y′ sub-phase spaces after four successive collimators (z¼0, 90, 180 and 270 cm) with and without collimators. The red dotregions indicate the transmitted particles and the blue dot regions are the scraped particles which are dumped onto the collimators. The green lines represent four successivefour-jaw slits. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

C. Xiao et al. / Nuclear Instruments and Methods in Physics Research A 738 (2014) 167–176 171

The magnetic quadrupole collimation channel has three cellsincluding three identical magnetic quadrupole doublets and fourfour-jaw slits. For efficient and flexible collimation, each cell is setto cause a phase space rotation of 451 and multiple cells withoverall phase advance rotation larger than 901 (1451 in thischannel). Four successive four-jaw slits are used for multi-stagecollimation with the phase space rotation in between.

The transverse acceptances of the quadrupole collimationchannel are calculated as follows:

Ax ¼h2

βx

; y¼ v2

βy

ð14Þ

here, h and v are horizontal and vertical half slit widths and are setthe constant for all slits. Utilization of collimation limits the beamemittance at the expense of decreasing transmission efficiency,and using slits with smaller size will result in smaller beamacceptance. Particle distributions with (red dots) and without(blue dots) collimators after four successive collimators are shownin Fig. 6. Since the quadrupole collimation segment is a periodicstructure in each cell, the red dot regions after the last collimatorsindicate the x–x′ and y–y′ phase space acceptances of the quadru-pole collimation channel.

6. Front-to-end simulation

After charge-to-mass selection, two normal quadrupole doubletsare used to match the analyzed beam into the quadrupole collima-tion channel. The quadrupole collimation channel, which consists ofthree normal quadrupole doublets with the same gradient butalternating sign is used to carry out the multiple-stage phase spacerotation. The decoupling section comprises two normal quadrupoledoublets and one skew quadrupole triplet, and their gradients areoptimized by a numerical routine to remove the inter-plane correla-tions, thus minimizing the product of the horizontal and vertical rmsemittances, i.e. maximizing the brilliance B.

Front-to-end simulation of the required species beam can bedivided into two separated simulations. In the first case, the four-jaw slits are pulled out and all of the particles can pass through thequadrupole collimation channel. In the second case, the four-jawslits are pushed in and only particles within defined acceptanceare transmitted through the quadrupole channel. Multi-particle

trajectories of the required charge state O3þ16 from the ECR ion

source to the beam line exit with and without collimators areshown in Figs. 7 and 8.

In general, an ECR ion source beam possesses a large beam sizeand divergence. Therefore, higher order effects (aberrations)cannot be avoided inside the solenoid. If the particle deviatesfrom the center of the solenoid, it feels a non-linear force andthe non-linear force causes the rms emittances and eigen-emit-tances to grow. Once the beam enters the analyzing magnet thehorizontal rms emittance starts to increase gradually, but thevertical rms emittance is not changed (see Fig. 9). After chargeselection, if the collimators are not adopted to cut particles, therms emittances and eigen-emittances are almost constant untilthe first skew quadrupole. Inside of the skew quadrupole triplet,rms emittances are made equal to the separated eigen-emittances.If the collimators are adopted, utilization of collimators stepwisedecreases the rms emittances and eigen-emittances in the match-ing section. After multi-stage collimation with phase space rota-tion in between, the beam is still coupled.

Fig. 9 illustrates the evolutions of the rms emittances andeigen-emittances along the beam line. Final four-dimensionalphase space particle distributions with and without collimators

Fig. 7. Multi-particle trajectories along the beam line without four successive slitcollimators.

Fig. 8. Multi-particle trajectories along the beam line with four successive slitcollimators.

Fig. 9. Rms and eigen-emittances evolutions along the beam line without (upper)and with (lower) four successive slit collimators.

C. Xiao et al. / Nuclear Instruments and Methods in Physics Research A 738 (2014) 167–176172

are shown in Figs. 10 and 11. In both cases, the inter-planecorrelation can be removed, and the beam quality in the secondcase is definitely better with respect to the first case.

If collimators are not applied, the y–y′ phase space plot lookslike a star (three-wing structure, 1201 symmetry). If the collimatorsare applied, the y–y′ phase space size is reduced nearly by a factorfour. In here, we define the initial dimensionless brilliance B to be1.0. The behaviors of dimensionless brilliances and the transmissionefficiencies along beam line with (horizontal half slit width h¼2 cmand vertical half slit width v¼2 cm) and without collimators areshown in Fig. 12. The transmission efficiency decreases from 100.0%to 47.1% (red solid line) and the dimensionless brilliance increasesfrom 1.00 to 9.91 (blue solid line) when collimators are applied. Ifcollimators are not applied, the transmission efficiency is still 100%(red dash line) at the exit of beam line and the dimensionlessbrilliance increases from 1.00 to 2.77 (blue dash line) after the inter-plane correlations are removed. By varying the aperture size of all

slits simultaneously with the same initial matched beam, thegradients of two normal quadrupole doublets and one skewquadrupole triplet are optimized to decouple the beam aftercollimation with different slit widths. The relationships betweenthe half slit widths and the transmission efficiency and dimension-less brilliance at the exit of beam line are shown in Fig. 13.

7. Measurement of coupling

The distribution of particles in phase space characterizes thebeam, and knowledge of the distribution of the particles in a beamis needed for accelerator design and for phase space manipulation.However, most of the published works in this area are onmeasurement of separated two-dimensional x–x′, y–y′ or l–δsub-phase spaces. There is some work on the measurement ofthe full four-dimensional phase space using a pepper pot

Fig. 10. Four-dimensional phase space distribution at the exit of beam line without four successive slit collimators. Horizontal and vertical rms Twiss parameters areindicated as well as the projected rms emittances.

C. Xiao et al. / Nuclear Instruments and Methods in Physics Research A 738 (2014) 167–176 173

technique [18], and these include measurement of full beam sigmamatrix and Twiss parameters. In this section, we refer to a solutionfor measuring the inter-plane correlation parameters using the fullfour-dimensional phase space reconstruction method. Thismethod is a special case of the full four-dimensional tomographyconcept proposed recently in [19].

The transport M of single particle coordinates from a position1 to a position 2 is

x

x′y

y′

266664

3777752

¼

m11 m12 m13 m14

m21 m22 m23 m24

m31 m32 m33 m34

m41 m42 m43 m44

26664

37775

x

x′y

y′

266664

3777751

: ð15Þ

Position 1 is the location where we seek to know the inter-plane correlation parameters ⟨xy⟩1, ⟨x′y⟩1, ⟨xy′⟩1 and ⟨x′y′⟩1. Sup-pose that further along the beam line at position 2 is a scintillating

screen on which the beam hits, and the scintillating screenmeasures the parameter ⟨xy⟩2. If the transport matrix M has thespecific form to cause the coordinates of a particle at position 2is related to a particle at position 1 by the following:

x

x′y

y′

266664

3777752

¼

μ1 0 0 00 1

μ10 0

0 0 ν1 00 0 0 1

ν1

266664

377775

x

x′y

y′

266664

3777751

ð16Þ

x

x′yy′

266664

3777752

¼

μ2 0 0 00 1

μ20 0

0 0 0 ν20 0 � 1

ν20

266664

377775

x

x′yy′

266664

3777751

ð17Þ

Fig. 11. Four-dimensional phase space distribution at the exit of beam line with four successive slit collimators. Horizontal and vertical rms Twiss parameters are indicated aswell as the projected rms emittances.

C. Xiao et al. / Nuclear Instruments and Methods in Physics Research A 738 (2014) 167–176174

x

x′y

y′

266664

3777752

¼

0 μ3 0 0� 1

μ30 0 0

0 0 ν3 00 0 0 1

ν3

266664

377775

x

x′y

y′

266664

3777751

ð18Þ

x

x′y

y′

266664

3777752

¼

0 μ4 0 0� 1

μ40 0 0

0 0 0 ν40 0 � 1

ν40

266664

377775

x

x′y

y′

266664

3777751

ð19Þ

we get

⟨xy⟩1 ¼⟨xy⟩2μ1ν1

; ⟨xy′⟩1 ¼⟨xy⟩2μ2ν2

ð20Þ

⟨x′y⟩1 ¼⟨xy⟩2μ3ν3

; ⟨x′y′⟩1 ¼⟨xy⟩2μ4ν4

: ð21Þ

It is an expansion of the inter-plane correlations measurement.Applying the transport matrices M1–M4, the distributions in x–y,x–y′, x′–y and x′–y′ phase spaces at position 1 are then recon-structed into the distribution in x–y phase space at position 2. Tocarry out this numerical calculation, we choose the last collimatorto be position 1 and the beam line exits to be position 2. Accordingto the previous simulation, the beam after collimation is stillcoupled, and we should assume that this distribution has some

x–y correlations so as to highlight the usefulness of knowing thefull four-dimensional distribution. The skew quadrupole triplet isturned off and the two normal quadrupole doublets are adopted toperform the phase space reconstruction, i.e. to provide the trans-ports defined in Eqs. (16)–(19). A numerical procedure is devel-oped to optimize the quadrupole gradients to meet the transportmatrix forms from M1 to M4, and provide the reasonable horizon-tal and vertical beam envelopes (see Fig. 14) between position1 and position 2.

Imaging to full four-dimensional-scenario Eqs. (16)–(19) repre-sent the application of well-known matrices of telescopic quad-ruplet, and the corresponding analysis of the selection ofquadrupole gradients is given in [20].

8. Conclusion and outlook

Intense ion beams with small phase space occupation (highbrilliance) are mandatory to keep beam losses low in injectoraccelerators like those planned for FAIR for instance. The particledistributions at the exit of an ECR ion source in phase space arestrongly x–y coupled. The spatial distribution is compact in thebeam core, and gets sparse far away from the core (star shaped). Ifa quadrupole collimation channel is applied, high-emittanceparticles are dumped onto the collimators and the core particlesare retained. Using collimators with smaller slits results in smallerbeam emittance and lower transmission. Since the beam is stillinter-plane coupled after collimation, the product of rms emit-tances is larger than the product of the eigen-emittances. In orderto further improve the beam brilliance, a skew quadrupoledecoupling section is placed downstream the quadrupole collima-tion channel. The inter-plane correlation which is created by thesolenoid field inside ECR ion source can be removed completely.Multi-particle simulations show that the combination of a quad-rupole collimation channel and a skew quadrupole decouplingsection improves the beam brilliance and transforms a correlatedround beam into a uncorrelated flat beam. The required four-dimensional diagnostics can be accomplished by application ofappropriate beam optics in combination with a scintillation screen.

Acknowledgments

This work is supported by the Helmholtz International Centerfor FAIR and the Bundesministerium für Bildung und Forschung.

Fig. 12. Dimensionless brilliance and transmission efficiency evolutions along thebeam line with (solid lines) and without (dash lines) four successive slit collima-tors. The horizontal and vertical half slit widths are 2 cm (For interpretation of thereferences to color in this figure caption, the reader is referred to the web version ofthis paper.).

Fig. 13. The relationship between half slit width and transmission efficiency anddimensionless brilliance. The skew quadrupole triplet is turned on to remove inter-plane correlations.

Fig. 14. The beam envelopes along the decoupling section for matrices M1–M4, seeEqs. (16)–(19). Only the normal quadrupole doublets are used.

C. Xiao et al. / Nuclear Instruments and Methods in Physics Research A 738 (2014) 167–176 175

One of the authors, Chen Xiao, would like to express his sincerethanks to Yuri Batygin at LANL, Youjin Yuan at IMP, Ludwig Dahland Stepan Yaramyshev at GSI for their fruitful discussions andgreat helps.

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