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This is to certify that the
dissertation entitled

BEAM PHYSICS DEVELOPMENTS FOR
A RARE ISOTOPE ACCELERATOR

presented by

Mauricio Portillo

has been accepted towards fulfillment
of the requirements for

._Eh.D.__degree ill—Blush:—

«7724/,

Major Professor

 

Date 9(/2? ’5

 

MSU is an Aflirmau‘ve Action/Equal Opportunity Institution 0- I 2771

 

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6/01 cJCIRC/DateDuepBS-p. 15

-__——_

' BEAM PHYSICS DEVELOPMENTS FOR A RARE
ISOTOPE ACCELERATOR

By

Mauricio Portillo

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Physics and Astronomy Department

2002

ABSTRACT

BEAM PHYSICS DEVELOPMENTS FOR THE RARE ISOTOPE ACCELERATOR

By

Mauricio Portillo

In support of a proposal for a Rare Isotope Accelerator facility, this thesis
provides a preliminary analysis of a number of related subsystems. An overview
of the requirements for the driver accelerator, production stations, and beam puri-
fication systems is presented. Some minor developments in the theory of beam
transport and acceleration are presented in order to discuss a technique for isobar
separation and multiple charge state selection. Changes to the COSY INFINITY
code for carrying out map-based calculations are described. The results obtained
by simulation are presented in detail for an isobar separator and a multiple charge
state selection system. The concept of beam stripping is discussed in order to
characterize the components of the multiple charge state beams. The production
of rare isotopes via spallation of heavy targets using fast protons is discussed.
Results obtained from experiments at an ISOL facility with direct and two-step
target geometries are presented. Implications of the results to the design of future
targets for rare isotope production are included. Some developments in beam
diagnostic techniques are discussed along with the experimental results obtained

from them.

Copyright by
Mauricio Portillo
2002

To all my family and friends who have listened to my ideas and cared for my well-

iv

ACKNOWLEDGMENTS

This work was supported by the US. Department of Energy, Nuclear Physics
Division, under Contract W-3 l-lO9-ENG-38 and by Physics & Astronomy Department,
Michigan State University. Special thanks go to my committee members: Jerry Nolen,
Martin Berz, Brad Sherrill , Wayne Repko, Vladimir Zelevinsky, Norman Birge.

Thanks go out to the following friends, colleagues, and relatives for their care and
support: Leslie Buck, Renee Irwin and Jan, Petr Ostroumov, Robert Janssens, Ben
Zeidman, Vladislav Aseev, Declan Mulhall, Teresa Barlow, Maria Grah, Wei-Qi Shen,
Art Ruthenberg, Frank Nelson, Simanta ‘Chintu’ Das, Naomi Leach, Todd Brown, as
well as many others not mentioned.

To my relatives who believed in me and accepted me for the person that I am and
the choices I made. Most importantly, I thank my mother, father, and brother for giving

me their support and strength to overcome many obstacles.

TABLE OF CONTENTS

LIST OF TABLES ............................................................................. xi
LIST OF FIGURES .......................................................................... xii
1 INTRODUCTION ....................................................................................................... l
1.1 The need for beams of rare isotopes ...................................................................... l
1.2 Technological aspects of the RIA driver accelerator ............................................. 6
1.3 Overview of contents ............................................................................................. 8
1.4 Beam transport concepts and notation ................................................................... 9
1.4.1 The reference particle .................................................................................. l 1
1.4.2 Equations of motion in relative particle coordinates .................................. 12
1.4.3 The transfer map and characteristic symmetries ......................................... 14
1.4.4 Focusing properties as determined from the first order map elements
19
1.5 Numerical approach to computing rays ................................................................. 22
1.5.1 Ray trace method of computation ............................................................... 22
1.5.2 Computation of maps .................................................................................. 25
1.6 Thephasespaceofbearns ..................................................................................... 31
1.6.1 The concept of the phase space ellipse ....................................................... 32
1.6.2 The normalized emittance ........................................................................... 34
1.6.3 The transformation of the sigma matrix ...................................................... 36
1.6.4 The rms emittance and higher order efi‘ects ................................................ 41
2 DESIGN OF AN ISOBAR SEPARATOR USING AXIALLY SYMMETRIC
ELEMENTS WITH ACCELERATION ..................................................................... 45
2.1 Isobar'ic purity and mass separators ....................................................................... 45
2.2 Achromatic mass separators .................................................................................. 48
2.2.1 Double-focusing spectrometer .................................................................... 50
2.2.2 Dual-potential spectrometer ........................................................................ 51
2.3 Correcting higher order abenations ....................................................................... 58
2.3.1 Aberrations at magnetic sector sections ...................................................... 59
2.3.2 Obtaining homogeneous sector fields ......................................................... 62
2.3.3 The deceleration column ............................................................................. 67
2.4 Purity according to the enhancement factor .......................................................... 73
2.5 Issues related to beam matching and the pre-separator ......................................... 79
2.5.1 Obtaining the required aspect ratio ............................................................. 80
2.5.2 Choosing the scheme of separation ............................................................. 82
2.5.3 Considering some aspect of the pre-separator ............................................ 85

APPLICATIONS WITH ACCELERATING RF DEVICES OF AXIAL
SYMMETRY .............................................................................................................. 88

3.1 Axially symmetric devices with time-varying fields ............................................. 88

 

3.1.1 Time-varying fields in COSY ...................................................................... 89
3.1.2 Properties of the longitudinal phase space .................................................. 91
3.1.3 Map calculations with RF devices ............................................................... 95
3.2 Applications with multi-q beams .......................................................................... 102
3.2.1 The conditions for an isopath ...................................................................... 105
3.2.2 Transporting and filtering of multi-q charge state beams ............................ 110
3.2.2.1 The 180° bend rebunching q-state filter ......................................... l 1 1
3.2.2.2 Phase space calculations ................................................................ 115
3.2.2.3 Other isopath systems under consideration .................................... 120
3.3 Determining the distribution of q-states ................................................................ 124
3.3.1 Importance ofmaximizing q ........................................................................ 124
3.3.2 The charge state evolution process .............................................................. 126
3.3.3 Empirical methods for determining F(q) ..................................................... 132
3.3.4 Codes available for determining charge state evolution .............................. 138
3.3.4.1 ETACHA ....................................................................................... 138
3.3.4.2 GLOBAL ........................................................................................ 140
3.3.5 Estimating q-state distributions for the RIA driver ..................................... 141
3.3.6 A final word about target survival and beam quality .................................. 148
PRODUCTION OF NEUTRON-RICH ISOTOPES BY A TWO-STEP
PROCESS ................................................................................................................... 152
4.1 Motivation and applied approach ......................................................................... 152
4.2 Applying models of production ............................................................................. 155
4.3 Target and extraction system ................................................................................. 156
4.3.1 Production of U0: materials ...................................................................... 156
4.3.2 Target layout ............................................................................................... 157
4.4 Isotope transport and detection .............................................................................. 160
4.4.1 Measuring the transported products ............................................................ 161
4.4.2 Product release fi'om target ......................................................................... 164
4.5 Models of the delay function ................................................................................. 167
4.5.1 Difi‘usion-Effusion based release model ..................................................... 167
4.5.2 Simplified analytical model of release ........................................................ 172
4.6 Comparison of the two production methods ......................................................... 176
4.7 Purity of beams ...................................................................................................... 178
4.7.1 Time dependent decay measurements .................. .- ............... 179
4.7.2 Absolute yields ............................................................................................ 181
4.8 Mixture targets ...................................................................................................... 185
4.9 Summary ................................................................................................................ 187
DEVELOPMENTS IN BEAM DIAGNOSTICS ........................................................ 189
5.1 Design features ofthe BIM ................................................................................... 189
52 Low-background ion counting system ................................................................... 194
5.3 Beam emittance with wire scanner ......................................................................... 197
5.4 Properties of secondary electrons and effects on detector ...................................... 198
5.5 Measuring transverse density distributions ............................................................. 204

vii

5.6 Emittance profile measurements with BIM ............................................................ 206

 

5.7 Summary ................................................................................................................. 212
6 DEVELOPING EXPERIMENTAL TECHNIQUES FOR RIA ................................. 214
6.1 Adapting the Dynarnitron accelerator facility for RIA developments ................... 214
6.1.1 About the accelerator .................................................................................. 214
6.1.2 Experimental beam lines ............................................................................ 217
6.2 Source development - _ .................................................................. 220
6.3 Mass separator and ion source performance .......................................................... 224
6.3.1 Emittance of source .................................................................................... 226
6.3.2 Mass separation characteristics .................................................................. 227
6.4 Isotope production with neutrons .......................................................................... 232
6.4.1 Z dependence of neutron production ......................................................... 232
6.4.2 Estimating the production by analytical models ......................................... 236
6.5 Recommendations for future studies ..................................................................... 242
6.5.1 Enhancements to the accelerator ................................................................ 242
6.5.2 Enhancements to source and detection performance .................................. 243
7 SUMMARY ................................................................................................................ 247
APPENDIX ....................................................................................... 250
A CANONICAL TRANSFORMATIONS AND THE SYMPLECTIC
CONDITION ............................................................................................................... 251
B SPECIAL VERSION OF COSY INFINITY 8 ........................................................... 258
8.] Changing reference particle energy ....................................................................... 258
8.2 Particle optical elements ........................................................................................ 259
32.1 Analytical function generated models .......................................................... 259
82.2 Modeling structures with charged multi-rings ............................................. 260
82.3 Other procedures with and without time-varying fielcb .............................. 263
82.4 Accelerating columns ................................................................................... 264
C OBTAINING VALUES FOR THERMAL IONIZATION EFFICIENCY ................. 272
D FORM OF SOLUTION OF THE TIME-RELEASE RATE EQUATION ................. 274
BIBLIOGRAPHY .............................................................................. 277

viii

2.1
2.2
3.1
3.2
3.3
3.4

4.1

4.2
6.1

LIST OF TABLES

List of coefficients determined by numerical simulation ........................................ 70
List of electric quad excitation voltage values in kilovolts. .................................... 81
Parameters used for TWOGAP cavity in Fig. 3.1. .................................................. 96
Properties of the incoming beam for the model described by Fig. 3.1. .................. 97
Parameters used for the elements in Fig. 3.8. ......................................................... 114
List of values that compare the results from the ETACHA code and analytical

codes for stripping at 9.43 MeV/u. Thickness, D, is in mg/cmz. ........................... 145
Parameters that give the best fit to the delay release curves plotted in Figure

4.4. .......................................................................................................................... 169
Estimated fractional contamination for Rb(a) and Cs(b). ....................................... 183
List of materials expected to be exposed to d beams .............................................. 233

1.1

1.2

1.3

1.4

1.5

1.6
1.7

1.8

2.1

2.2

2.3

LIST OF FIGURES

Diagram illustrating the production and transfer method in a thick stationary
target. ..................................................................................................................... 6

Scheme for the Rare Isotope Accelerator system at Argonne. The regions
labeled with Exp represent points at which certain experiments can make use
of the products. ....................................................................................................... 7

Schematic representation of several focusing conditions in which elements of

the map may vanish. (a) Represents parallel-to-point focus in which (x,x)=0.

(b) Represents point-to-parallel focus in which (a,a)=0. (c) Represents
point-to-point focus in which (x,a)=0. (d) Represents parallel-to-parallel

focus in which (a,x)=0 ............................................................................................ 21

Enge functions for dipole and quadrupole elements as a function of position
over gap width, G. The Enge coefficients used were derived from

measurements as cited in the text. .......................................................................... 31

Diagram illustrating the points of extremum in the phase space ellipse along
the x—a plane. The tilt is such that a,<0 ............................................................... 34

The evolution of the x—a phase ellipse through two equivalent drifts ................... 38

Sextet system of total length L rmde up of electric quadrupoles as described in

the text. Plots of the ray trajectories along the x—s and y—s planes are shown

in the top two plots, while snap shots of the x—a phase space ellipse are

shown for both a first order (line) and a fifth order calculation (dots). Other
setting are as follows: quad apertures radius = 2.5 cm, quad length =15 cm,
L1=30cm,L2=20cm. ........................................................................................ 39

Plots of three differing phase space distributions with equivalent rms

emittance. The type of distribution is labeled above each plot and at the lefi

side of each plot is the percentage of particles lying within the corresponding
ellipse of emittance ex. ............................................................................................ 43

Double-focusing spectrometer with rays of multiple divergence as well as
multiple energies. Rays are focused in the horizontal plane but quit by

varying energy at point B. ....................................................................................... 50

Layout of dual-potential spectrometer. The spectrometer is broken down into
4 sections as described in the text [PortilloOla]. Overall footprint is 20m x 30
m ............................................................................................................................. 52

Mass spectra in x—a phase space for three masses of similar boundary-type
initial distributions. Going from the top-left to top-right plot shows the effect

2.4

2.5

2.6

2.7

2.8

2.9

on the mass spectrum at the end of section H when adding a random energy

spread that lies between 6K=-A and +A. The dual-potential separation results

in the spectrum taken at the end of section L, which shows the effect of the
achromatic correction .............................................................................................. 55

Beam envelopes for the dual-potential spectrometer in the horizontal and

vertical planes. Plots are illustrated for the s—y (a) and s—x (b) planes for the
effects of the transverse phase space on the rays. The last plot illustrates the

effect of energy dispersion along x. ........................................................................ 57

Effect of applying each successive multipole field on the x—a phase space
plot calculated to 5th order at the end of section H. The effects fi'om the y—
phase space are included ......................................................................................... 59

Top view of a magnetic sector in which positive edge angles have been

imposed at both entrance and exit positions. In addition, a round curvature

shape (small dashes) has been cut out at the exit to correct the second order
aberration. The simple sector (thin line) is that of the sector before edge

angles and curved edges are imposed. .................................................................... 61

Effect of turning on the hexapole field correction on the y—b phase space at
the end of section H. ............................................................................................... 62

Diagram showing the lines of magnetic flux for the design of the DH sector of
section H. The beam occupies a region that extends up to 43 cm in the
horizontal. ............................................................................................................... 63

Plots of the y—component of the B-field versus the x—position fi'om the
center of the DH sector. The shape H-magnet is varied in depth in an attempt
to make the distribution more uniform. .................................................................. 64

2.10 Plot of the x—a for a 5th order calculation of a magnetic sector with

2.11

2.12

2.13

homogeneous field distribution and another with the field distribution
predicted by POISSON. .......................................................................................... 65

Diagram showing the lines of magnetic flux for the design of the DL sector of
section L. The beam is expected to occupy a region that extends up to about
16 cm in the horizontal. .......................................................................................... 66

Immersion lens structure with ions coming in with parallel trajectories from
left to right. The lines of potential along the center illustrate the fringing
obtained in the gap region. ..................................................................................... 69

Plot of the final versus initial radial positions, r, and n, respectively, of the
simulation in Figure 2.12. A 5th order polynomial fit is used to extract the
aberration coefi'rcients from Simion 7 results. The curve obtained from

applying the aberrations according to COSY are plotted for comparison. The
coefficients are listed in Table 2.1. ......................................................................... 70

xi

2.14 Cross section of the deceleration column along the x—s plane of section I.
The beam comes in from the left at 100 keV then gets decelerated to 29.9 keV
in the first gap and to 10 keV in the second. ........................................................... 72

2.15 Plot of beam envelopes along x and y. The dotted line represents the center
position of the gap for the labeled amount of deceleration. .................................... 72

2.16 Phase space distributions in the x—a and y—b phase space given by Simion
and COSY. .............................................................................................. 73

 

2.17 The transmission and enhancement factor as resolution of R...=20,000 as a
function of the order of the calculation. .................................................................. 75

2.18 (a) Concentrations of wanted and unwanted species afier separation as a
function of resolving power. (b) EF as a function of resolving power. Inset
plot shows the separation of two masses according to R... ..................................... 77

2.19 Mass spectra at two sections of the mass separator for WM =20,000. The top
most figure (a) illustrates the spectrum for a beam with no energy spread at
the focal plane of the first section of the separation. The next plot (b) shows
what a Gaussian distribution in energy with 95% of particles having energy
between AK/K=:l:5x10'5 (21:5 eV at 100 keV). The last plot (c) demonstrates
how the achromatic character added by the section after deceleration can
improve the resolution. ........................................................................................... 79

2.20 Sextet with magnifications of M, =-1/J8 and My=-1. Dimensions are similar
to those of the sextet in Fig. 1.7, except that the triplets have been shifted
outwards fromcenterby11.5cm ............................................................................ 81

2.21 Various schemes of source extraction, separation, and post-acceleration.
Preseparation and beam matching have been left out for simplicity. ..................... 83

2.22 Conceptual design of preseparator for the RIA facility. The beam must be
transported from below ground level to the isobar separator and rare isotope
accelerator above ground. ....................................................................................... 86

3.1 Two-gap structure system symmetric about the center. .......................................... 95

3.2 Relative phase dependence of the quantities (1,6,0, (6&1), l/f, and (Kay-Kopy
K09. ......................................................................................................................... 99

3.3 Cavity potential V0 necessary to obtain upright ellipse at exit of system given
length L before the first gap. ................................................................................... 101

3.4 Resulting phase space plots for each corresponding r, as evaluated to 5th order
by COSY. ................................................................................................................ 102

xii

3.5

3.6

3.7

3.8

3.9
3.10

3.11

3.12

3.13

3.14
3.15
3.16
3.17

3.18

3.19

Simplified layout of RIA driver linac under the acceleration scheme of
uranium. SRF signifies array of superconducting cavity structures ....................... 103

Path of the reference particle and some arbitrary particle of length Lo and L,
respectively. ............................................................................................................ 107

Simplest possible pure magnetic achromatic systems. The possibility of

obtaining an isopath by the mirror symmetric system (a1) and four dipole

system (bl) is determinable from their respective dispersion functions, (a2)

and (b2), as described in the text. ........................................................................... 108

Diagram illustrating the 180° bend rebunching q-state filter. The system is
mirror symmetric about point 7. The first order longitudinal phase space
ellipses are shown for two q-states, one of which has charge qo and the other

 

qo+Aq. The details of the beam dynamics are described in the text ....................... 113
Plot of beam distribution for position 5 in Fig. 3.8 ................................................. 115
Transverse phase space plots at point 3 in "fig. 3. 8. Calculated with COSY to

2nd order. ................................... - ................................................. l 16
Transverse phase space plots at point 3 in Fig. 3.8. Calculated with COSY to

2nd order (top pair) and 3rd order (bottom pair). ................................................... l 17
Longitudinal phase space ellipse at exit of 180° bend system. Calculated to

3rd order with COSY. Boundary of all q-states overlap. Dashed boundary

ellipse contains 50% more area than the bounded phase space area. ..................... 118
Diagram of an alternative scheme of obtaining an isopath system with four

dipoles and without the use of a chicane as in the system in Fig. 3.8 does

(left). Plot of the dispersion function along the optic axis (right). ......................... 120
180° Bend scheme. .................................................................................................. 121
Dog-leg scheme. The SRF stands location for superconducting RF cavities. ....... 123
Chicane scheme. ..................................................................................................... 123
Hypothetical quantum state level diagrams showing a simplified transition

from a q to q+l ionization state. ............................................................................. 127

Charge state evolution of 238U on Al foil according to ETACHA. A
comparison is made between a ealculation using cross sections that are
corrected for energy loss (top) and one without corrections (bottom). ................... 131

Equilibrium thickness for uranium bms according to formulation by Baron
and Dmitriev as explained in the text. Also shown are values determined as
explained in the text. ............................................................................................... 136

xiii

3.20 Evolution of charge states for K/A=45 MeV/u ions of U according to the

3.21

3.22

3.23

3.24

4.1

4.2

4.3

4.4

4.5

ETACHA code. The maximum mean charge occurs at ~21 mg/cm2 at which
point (7 =52.5 and 5 =0.69 ..................................................................................... 137

Mean charge states evaluated from charge state evolutions of 24.1 MeV/u
238U from experimentally measured values [Leon98] and from the code
ETACHA. The lines that intersect mark equilibrium thickness (0.7 mg/cmz)

and mean charge ( (7 =76.6) according to the analytical models. ............................ 140
Evolution of charge states at the high energy range of uranium on carbon as
calculated by GLOBAL. ......................................................................................... 142
Charge state evolution according to ETACHA for 238U going through

various types of foils. Cross sections corrected for energy loss ............................. 145
Charge state evolutions according to ETACHA and GLOBAL calculations of

uranium under Li and C foils. No corrections for energy loss done for the
ETACHA results. .................................................................................................... 147

Cross sectional top view of the target/source used for rare isotope production.

All objects are cylindrical except for the Ta block, which has a square cross

section in die beam direction and rectangular as viewed from the top.

Secondary neutrons produced by the 1 GeV proton beam on the Ta target

react with the 238U in the target to produce the isotopes of interest ...................... 158

Diagram describing the tape transport system used for collection isotopes as

ions focused through the slit at the image plane. Two solid-state detectors

were used for detection of nuclear decay while a Faraday cup and electron
multiplier setup is used to measure ions directly. ................................................... 162

Ionization efficiencies for a tungsten tube at 2800 K according to the surface
ionization (3,.) and hot cavity thermal ionization (8”) models .............................. 163

Time dependent release of (a) 88Rb and (b) 139Cs isotopes from the

target/source after issuing a pulse to the target in direct production mode. The
average temperature of the target is about 2500°C. Two models are applied

for comparison by way of a chi-square minimization algorithm to arrive at the
relevant parameters that yield the best fit. The curves are labeled for the
diffusion/effusion model (D+E) and the analytical (an) equation model

described in the text. ............................................................................................... 170

Release coefficients obtained from applying the diffusion/effusion model

(D+E) and the simplified analytical (an) form of the release curves. The

release curves of Rb exhibit a faster, more efficient release process than the

curves of Cs. The Rb curves given by the two models overlap with each

other, while the Cs curves have a well-marked difference. .................................... 173

xiv

4.6

4.7

4.8

4.9

(a) Mass yield distribution of the Rb isotopes in the indirect target

configuration when striking the neutron generator with 1 GeV protons. A

curve is given for the yields given by the LAI-IET code for [>20 MeV

particles as well as those resulting fi'om the production by neutrons of E<20

MeV. (b) Similar mass distribution for the Cs isotopes. ....................................... 175

(a) Plot of the ratios for the measured yields in the direct and indirect

configuration in the neutron rich Rb mass region. Plotted also are the

predicted ratios given by the Monte Carlo calculations. (b) Similar plot for

the neutron rich Cs mass region. ............................................................................. 177

Measured activity as a function of time at the solid state detector (det-l on
Figure 4.2) at various mass regions. The various line graphs result from
simulation as described in the text. ......................................................................... 180

Experimentally measured values of the absolute yield at the target obtained by
correcting the values of R for release, ionization, and transport efficiency.
The production for Cs isotopes as given by the Monte Carlo models is shown

for comparison. ....................................................................................................... 181

4.10 Direct to indirect production ratios as calculated by the Monte Carlo models

4.11

5.1
5.2

5.3

5.4
5.5

5.6

for Cs, Ba, La, and Ce. Relative to Cs, the Ba ratios are very similar but a
significant difference is observed; especially for Ce. ............................................. 184

Fast neutron induce fission rate as a function of the occupied volume of the

neutron generator. The neutron generator here is taken to be tungsten mixed

with the UCx fission target of 3.1 g/cm3. Using a mixture with 62% UCx

powder gives an enhanced fission rate by ~60% relative to the rate at pure

UCx (VJV=1). ......................................................................................................... 187

Diagram of the BIM ................................................................................................ 190

Diagram SE collection region of BIM. Trajectories of electrons exiting the
surface at angles ranging from -60° to 60° with a kinetic energy of 25eV are
simulated with Simon using a 330 um spaced grid. Bias potential used for
each element is shown along the bottom. The beam coordinate system is

oriented such that the z-axis points in the direction that the beam travels .............. 191
Ion counting detection system based on a channeltron electron multiplier and

a BeO conversion surface. Resistance is in units of Ohms. ................................... 195
Detection efficiency for counting ions of 133Cs+1. ............................................... 196
Lay-out of the 8°+27° West beam line at Dynarnitron where the BIM was

tested. ...................................................................................................................... 196
Diagram illustrating the slit/wire emittance scanning system ................................. 198

XV

5.7 Simulation of energy distribution (bar graph) compared to the experimental

5.8

5.9

data of Rothard as explained in the text. The parameters used with the
probability density function are listed. .................................................................... 201

(a) Broadening of the secondaries as they are transported to the MCP for

detection as calculated by Simion 7. A 13 kV potential was used on grids of
different mesh size. The ms of the transverse velocity, 6v, component

increases with mesh size, which causes a; to increase almost proportionately.

(b) Results of simulation showing the a. (left axis) and 6v (right axis)

dependence on grid potential for a 0.33 mm grid. The solid line is evaluated

from an equation that explains what the physical significance of the process

where 6v is taken to be a linear function of the grid potential. ............................... 203

Beam images of a) low intensity ~4><102 pps l8 keV/u krypton beam, area

size is 5.1 mm x 6.5 mm; b) Total intensity of 2.5 X105 pps of a radioactive

beam line. Area covered in coordinate system of the beam is ~17.0 mmx18.2

mm. ......................................................................................................................... 205

5.10 Beam profiles in the y-direction measured by both the wire scanner and BIM

systems when the grid accelerating potential is 13 kV. .......................................... 206

5.11 Double slit plate used to scan both the x— and y-profiles. The orientation is

such that the beam is coming out of the page and the relative beam coordinate
axes are labeled on the bottom left. ........................................................................ 207

5.12 Emittance profiles extracted from the divergence profiles as described in the

text. The ellipse drawn fits the phase space area given by 8, and 8,. If the
distributions were truly Gaussian in form then 90% of the total beam intensity
would lie within this boundary. The Twiss parameters and rms emittance

were evaluated by a statistical analysis. .................................................................. 210

5.13 Phase space profile measured with wire system. .................................................... 212

6.1

Scanned diagrams of original figures documented for the last modifications
made to the column of the Dynamitron accelerator. (a) is at the HV end by
the ion source and (b) is at the extraction end section. Notice the 1.5 inch

apertures that had been installed. ............................................................................ 215
6.2 Simulated beam envelopes for an accelerating column under different

accelerating gradients. The inset plot has the same horizontal scale and shows

the rays calculated for the 20 kV/gap case. ............................................................. 216
6.3 RFQ and target/source layout at the Dynarnitron accelerator facility. .................... 218
6.4 Diagram of surface ionization source for production target. .................................. 221

xvi

6.5

6.6

6.7

6.8

6.9

Beam current as a function of temperature for a Na and K from ionization at a
W%26Re wire. ........................................................................................................ 222

Filament power used to get cavity temperature under the following conditions:
1) Graphite at front, Ta shields in back 2) Zirconia type FBD at front, Ta
shields at front 3) Zirconia type FBD at front and back. ........................................ 223

Total current at each mass obtained with only power at the W%26Re ionizer
filament. .................................................................................................................. 225

Emittance area (top) evaluated from the emittance profile (bottom) measured
right before the entrance of the mass separator dipole ............................................ 226

Detailed mass spectrum between mass 22 and 42. The natural abundances
have been labeled next to the corresponding isotope peaks of Na and K. .............. 228

6.10 Simulation of beam energy suppressor. The 20 cm diameter tube is gridded at

6.11

6.12

6.13

6.14

the end go provide a uniform potential for suppressing particles of energy
below the applied potential. .................................................................................... 229

Mass region between Na and K detected with ion counting system for a beam
accelerated with an 8.5 kV potential. Peak appearing at mass 27 is likely that

of Al” (V,=5.99 eV). The broad peak at about mass 32 totally disappears

when the suppressor is above 8 kV. ........................................................................ 230

Calculated 1st order resolving power as a function of the potential at the first

lens. A mass dispersion of 1.4 cm/%m/q is estimated. The actual

measurements indicate a resolving power greater than the ealculations by

~22%. The limit occurs at about Am/m~600, at which point the acceptance

begins to suffer ........................................................................................................ 231

Diagram of target ladder set-up for neutron flux measurements of various
targets. ..................................................................................................................... 232

Count rate at neutron detector generated by deuteron beams colliding with
various materials at kinetic energy, K4 .................................................................... 234

6.15 Neutron yield, Y,, fi'om 3.4 MeV deuterons on a Be target (left axis) and

6.16

6.17

6.18

reaction cross sections by neutron of energy K. for the reactions listed in the
legend (right axis). .................................................................................................. 238

Production of certain isotopes versus the deuteron beam energy on a Be target. ...239

Mass spectrum at the mass 24 region with a 4.5 uA deuteron beam at 3.5
MeV on a 118 target. .............................................................................................. 241

Annular target configuration for increasing production efficiency at
Dynarnitron. ............................................................................................................ 245

xvii

Chapter 1

INTRODUCTION

1.1 The need for beams of rare isotopes

The concept of constructing a facility for carrying out advanced studies using
energetic beams of rare isotopes has been of interest for at least a decade [ISL92].
Interest has largely been aimed at developing a method that would provide beams of
short-lived nuclei having sufficient energy to break the Coulomb barrier. The beams
could be directed at thin targets to induce Coulomb excitation (testing collective
properties) and transfer reactions (testing single-particle aspects) of nuclei that have
otherwise been too difficult to handle because of their short half-lives.

An ideal secondary beam would be one having a normalized transverse emittance
$0.3 mm-mr and a longitudinal emittance of $5 keV/u-ns at energies ranging from 0.1 to
10 MeV/u. These types of conditions make it possible to obtain very short pulses of
beam with low transverse momentum spread to allow for adequate time-of-flight and
rigidity resolutions in nuclear studies. Current heavy ion accelerators are capable of
providing these types of beams by extracting the rare isotopes as ions from a standing
source. In recent years, however, there have been a number of facilities that have
extended the method to beams of rare isotopes [Nolen02a]. For the most part, such
efforts have been rather limited to a limited number of relatively light rare isotopes.
There is still much room left for improvement in terms of the yield, number of available

species, and efficiency of transport and acceleration.

Another technique that has been used to provide rare isotope beams is the in-flight
method [Sherrill92]. It requires that very fast nuclei passing through a thin target get
broken up by nuclear interactions at grazing incidence with light target nuclei. The
products essentially keep going with the same velocity as the incoming projectile and are
then directed to a secondary target. This requires that the secondary beam be purified
from the primary beam and other unwanted products by way of rigidity selection through
a fragment separator. One major advantage is that the secondary beam does not need any
further acceleration as it is delivered to the second target. The main drawback, however,
is that the secondary beam has a lot of transverse and longitudinal momentum spread
from the reaction at the primary target. To make the process efficient the fi'agment
separator must have a very large momentum acceptance and extra characterization of the
projectile before it reacts with the secondary target.

In recent years there have been on-going efforts to slow down the secondary beams
with a degrader after rigidity selection. The degrader is made of solid material and
should remove enough momentum from the beam such that it can later be stopped with a
gas filled trap [Savard99]. The trap, sometimes referred to as a gas (filled) catcher,
actually transports the ions with a DC field across a region that has confining RFQ fields.
During this transport process, the interacting with the gas cools them as ions until very
little momentum spread is left in the products. To keep the products ionized, the gas
should be filled with a very pure noble gas, such as helium. The cooled products are then
extracted at the other end of the trap for use in experiments.

Such a process is to be extended to providing cooled beams of rare isotopes for

subsequent acceleration in the RIA post accelerator. It is a very attractive approach since

the release times are expected to be down to as low as a few milliseconds and the process
is hrgely independent of the chemical properties of the products [Savard02]. Ion species
of refractory elements, such as tungsten and rhenium, should become accessible through
this technique.

The RIA facility will be based on the idea that the same driver accelerator that is used
to provide beams for fragmentation can also be used to produce rare isotopes from within
a stationary target [Savard01]. For the in-flight technique the beams usually consist of
the heaviest projectiles (Zp218) to collide with targets of the lightest possible species
(Z518). The target thicknesses must be relatively thin, such that nuclei traversing the
medium without reacting (primary beam) will loose smaller percentage of momentum
than the reaction products. For incident energies of 400 - 500 MeV/u the range of
thickness is expected to lie in the range of 1-10 g/cm2 [Jiang02]. The reactions that yield
most of the products can be grouped in a number of categories and are as follows:

0 Fission reactions occur when the collision induces enough excitation energy on
the projectile nucleus to cause it to de-excite by spontaneous fission. Particles as
light as a proton or deuteron at the target can induce these type of reactions.

0 Spallation occurs when a very direct impact between target and projectile occurs.
The high level of excitation can produce a wide array of products that are usually
not accessible through fission.

o Fragmentation reaction occurs when two nuclei collides at low impact parameter
with the target nucleus. One usually describes it as having the target nucleus

scrape off nuclear matter from the projectile as it passes.

The fragmentation process may be regarded as lying between the two extremes,
spallation and fission. At the extremes it is possible to distinguish the spallation and
fission processes through their respective mass distribution of the products. This point is
explained in more detail in Chapter 4. The important thing to keep in mind is that
fragmentation reactions are orchestrated through the combination of target and projectile
used in the reaction to obtain favorable kinematics from the reaction.

A critical component of the RIA facility will be the fragment separator. The
momentum acceptance of the device must be about i9% to deliver products at the gas
catcher and about 13% to deliver the products to secondary targets at the focal plane.
The optics requires that a beam, such as of uranium, be delivered at a 1 mm diameter to
the thin target. The power density in the foil is expected to get as high as l MW/cm3 and
has prompted the special development of windowless liquid lithium targets [Nolen02b].

Other techniques that are incorporated into the RIA plan will require production at
thick targets. This can be though of as a kinematically reversed situation of the in-flight
method since the beam will consist of light targets (ZPSI8) impinging on heavier ones
(2,252). The typical thickness of the target is expected to lie between 10 to 100 g/cmz,
such that the products are generated within the target matrix. The energy of the primary
beam per nucleon is usually higher since they are lighter species. The expected energies
lie between 500 to 1000 MeV/u. Most of the energy of the primary beam will be dumped
in the material, except for the last ~50 MeV/u, which produce a very small fraction of the
entire yield. The large fraction of energy that does react can produce a large flux of
neutrons from spallation reactions. A significant fraction of the neutrons have energies in

the few MeV range and will cause fission reactions on heavy nuclei, such as 238U.

Although fragmentation reactions occur, they are generally not mentioned in applications
with heavy targets since the products do not contribute beneficially as much as they for
do in the in-flight method.

The thick target must be kept at high temperatures (~2000°C) in order to enhance the
diffusion of the products out of the material. Target materials used must be refractory
and able to release products rapidly to minimize losses by decay. The exit port is
strategically positioned at the container to minimize the path length to the ionization and
extraction region. The diagram in Fig. 1.1 depicts the transport process from the target
matrix to the extraction region.

Notice that the figure indicates that the ionized products are extracted to a mass
separator. The process of purifying the products right after production has come to be
referred to as Isotope Separation On-Line, since the production and mass separation
occur continuously. The ions from the gas catcher are also expected to be purified with
the mass separator system. In that case, however, there will be no need for an ionizer
since the products remain ionized in the gas. Both processes are expected to be
complements of each other in the ISOL arrangement of the RIA facility as shown by the
diagram in Fig. 1.2. Both of these systems are to be able to provide beams of relatively
low momentum spread. Notice that the diagram indicates that the products from the in-
flight process are to be diverted to a direct experiment. This pertains to the secondary
target scenario where products of high momentum can react with a secondary target, as
mentioned earlier. Thus there are three techniques that are to become available with
beams fi‘om the RIA driver accelerator, which should give it a wide range of flexibility

for experiments with rare isotopes.

primary
beam

 

F
_ ‘ . to mass
. , ~. :’ Q; separator

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r- . . . .
$53 A \ IOI'IIZfltIOIT region] W
E extraction
r W
\ diffusion of

products

 

 

\

 

 

 

Figure 1.1
method in the on-line mass separation technique.

 

production target J

Diagram illustrating the production and transfer

Here, we shall only directly address the ISOL method through the production in thick

targets; hence, the ISOL method will always refer to the thick target scheme, unless

otherwise specified.

1.2 Technological aspect of the RIA driver accelerator

The productions that are necessary to sustain the RIA facility are at a level that

demands a high level of performance from the driver accelerator. The present design of

the driver requires that there be a total of 1.3 GV of effective RF acceleration available

for acceleration.

The beam power necessary is expected to be ~100 kW for the in-flight targets (high

2,) and ~400 kW to the thick targets (low Zp). The accelerator is designed to be a linac

 

 

 

 

 

 

 

/" Exp Exp

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Dnver Primary P199955 Mass
linac beam Rare isoto purification
. - Pc
Threk targets 1
Post-
acceleration
Minor aspects covered
a Strip/filter at driver
a Release from ISOL targets
a Mass separation
0 Matching beams into accelerating structures
0 Ion detection
0 Phase space diagnostics

 

that relies

maximize the energy of the ions over the entire spectrum of elements (lSZpS92) all
cavities must be independently phased. In the case of protons the expected final energy is
about 800 MeV at an output of 400 kW, where the limiting factor is expected to be the
amount of available RF power. At the other end of the spectrum, it is expected that up to

~100 kW of 400 MeV/u uranium ions can be delivered to in-flight targets. The limiting

factor here

two stripper foils. This problem has been addressed by implementing multiple charge

Figure 1.2 Scheme for the Rare Isotope Accelerator system
at Argonne. The regions labeled with Exp represent points at
which certain experiments can make use of the products.

almost exclusively on superconducting cavity structures [Shepard99].

is the efficiency of charge stripping of uranium ions at the source and at least

state acceleration as described in Chapter 3.

 

1.3 Overview of contents

This thesis addresses some of the concepts needed for the RIA facility. The pictorial
representation of the concepts in Fig. 1.2 should be seen as a macroscopic view of only a
handful of many challenging aspects of the facility. There are a number of ways to
approach the problem and studies continue to find which solutions are viable in terms of
cost and technological requirements. As seen by the diagram, a large part of the effort
requires beam development; hence, a lot of the topics covered will deal with beams.
Some knowledge related to topics in numerical simulation by ray tracing and mapping
are helpful to the reader, and there are a variety of books and other manuscripts that may
be used for reference. Inevitably, there are some details that need some closer attention;
therefore, a brief introduction related to beam theory is provided in the following sections
of this chapter. It serves as a reference for much of the notation and terminology that is
to be used for the remainder of this dissertation.

The chapter that follows will begin by demonstrating that an implementation to the
COSY INFINITY code system can allow for the simulation of beams that are accelerated
by electrostatic potentials. Under such conditions the energy of all the particles with
equivalent charge may change by the same amount, and the effects on the transverse
phase space must be simulated. It will be shown that it is possible to use higher order
maps to design a mass separator for the purification at the isobar mass level. The
separator is necessary for mass purification at the post-accelerator.

A chapter will also be devoted to the simulation of time-varying fields on beams with
realistic radio frequency bunch structure. The COSY INFINITY code system has been
modified to permit simulation of radio frequency cavities using higher order maps. A

method of transport maps will be applied by applying some further modifications to the

COSY INFINITY code system. It will be demonstrated that a bunched beam of multiple
charge states can be transported through a system of magnets and RF devices in order to
match the beam between sections of the driver linac.

The fourth chapter will be devoted to topics related to the production and release of
rare isotopes using a conceptual target set up at an ISOL facility. Although, the focus
was initially intended for comparing the production of neutron rich isotopes by using
either fast neutrons or relativistic protons, it is important to shed light on the aspects of
release time, ionization efficiency, and detection. It will be shown that all such topics
play a vital role of optimizing ISOL targets for facilities such as RIA.

The remaining chapters will be devoted reporting results that were found in these
studies that are relevant to the RIA facility. A number of beam tests from a surface
ionization source were carried out at a mass separator set up constructed at the
Dynamitron facility at Argonne National Laboratory. We report the emittance
measurements and production rates that were observed and make recommendations for
future studies. Some focus will also be shed on instrumentation that has been designed
for RIA applications, and has either been constructed and tested for the RIA facility or
are still under development. One of these instruments is used to image low intensity

beams using multichannel-plate technology.

1.4 Beam transport concepts and notation

In the study of beam transport, or beam optics, it is necessary to develop an approach
for determining the orbit of charged particles in the presence of electromagnetic fields. It
is necessary to derive the equations of motion that take into account any interaction that

will have a significant effect on the trajectory motion. There are a number of approaches

that may be adopted in deriving the equations of motion under electromagnetic
interactions. For example, the Lagrangian or Hamiltonian techniques may offer a more
direct window into the conservation laws by knowledge of the electromagnetic potential.
There is also the more common approach in which the concept of a force is applied in
deriving the equations of motion. The Lorentz equation in vector form states that the
electromagnetic force on a charged particle is given by
F=qe(E+va), (1.1)!

where q is the charge number and sign, while e is the unit of charge in SI units. The
particle moves with velocity, v, in the rest flame of a magnetic field, B, and electric field
E. Choosing a Cartesian frame of reference the form of the equations of motion can be

expressed in relativistic from as follows:

dP dR P/m

—= ——= v = (1.2)
dr dt J1+(P/mc)2

Here, m is the rest mass of the particle and P is its momentum. Depending on the form of
the electric and magnetic fields, much effort may be devoted to finding analytical forms
of the solution or writing algorithms to solve the differential equations numerically, or a
combination of both. On the other hand, a large part of the effort may also go into
determining the form of the electric and magnetic fields. One needs to consider the
amount of accuracy that is necessary for the application and detail about the geometry of
the elements that induce the fields. In some applications it is even necessary to account
for the self-interaction of particles in the beam, which is sometimes referred to as the
space charge effect. Evaluating the field distribution becomes even more complex when

considering the effects of time-varying fields. The effect of time-varying fields is left to

10

the Chapter 3, and here we only consider beams transported through electrostatic and

magnetostatic elements.

1.4.] The reference particle
Most structures used for optical elements in beam optics have very well characterized

properties and are designed with the intent of having some type of symmetry about the
motion of an ideal particle trajectory. This ideal particle, commonly referred to as the
reference particle, follows a particular trajectory path that is known from the properties of
the fields. For example, a homogeneous dipole is designed such that the reference
particle enters at a certain point, follows a circular path, and exits at a predestined
location.

In actuality, beams consist of particles that have trajectories close to those of the
reference particle but deviate within some proximity. Sources of charged particles are
designed to produce particles with properties very close to those of the reference particle,
but are limited to producing ensembles of particles whose coordinates are those of the
ideal reference particle only on the average. The deviation from this average is usually
quantified as the emittance in phase space, which in effect specifies the amount of
volume occupied by the ensemble of particles. The design of any beam transport system
should be tailored such that some specified acceptance is obtained and that the extent of
any deleterious effects on the emittance by the applied fields is kept to a minimum. Since
the cost of an optical system tends to go up with the amount of acceptance, it is
advantageous to select a particle source that produces beams of minimum emittance at
the amount of particle flux that is necessary. More on the beam phase space will be

discussed later.

11

The path and instantaneous momentum of the reference particle must be determined
for any beam optics problem. For most optical elements the path is well known.
Otherwise, the path needs to be determined by solving equations (1.1) and (1.2)

analytically or by numerical ray tracing methods within a stationary frame of reference.

1.4.2 Equations of motion in relative particle coordinates

In a stationary reference frame, the location of any arbitrary particle in the beam can

be specified in Cartesian coordinates as

R=(X.p..Y.p,.Z.p.), (1.3)
in which each variable is a function of time. If the position of the reference particle,

Ra =(Xo.po..Yo.po,.Zo.po.l. (1.4)
is known at any time, then we can define a relative coordinate system in which the
position of an arbitrary particle may be expressed by the position vector

r = R - R0 (1.5)

In this moving frame of reference, the direction of the z-axis is parallel to the momentum
vector of the reference particle. These set of relative coordinates form a set of canonical
conjugate pairs. By canonical transformations a more convenient set of coordinates can

be derived that can be used to describe the motion of beam particles. The following set

of canonically conjugate pairs is convenient to use in describing ensembles of particles

[Berz90a]:
rl=x=(X—Xo) r2=a=(px—p0x)lp0
r3=y=(Y-Yo) r4=b=(P,-Po,)/Po
r, =l=—(t-to)fl0 r6=6K =(K—Ko)/K0 (1.6)

12

Notice that the momentum variables in the transverse plane are now divided by the
magnitude of the momentum of the reference particle, p0, to obtain the variables a and b
in units of radians. The relative position variables x and y have units of meters. The

variable I is proportional to the difference in the time-of-flight between the particle of

interest and the reference particle by the time-position factor, #0 = var/(1+ y). yis the

total energy of the particle divided by mc2 and Va is the velocity of the reference particle.
From now on we treat m as the rest mass of any particle and mo as the rest mass of the
reference particle.

In Appendix A it is shown how the canonical transformation can be used to obtain 3
as the independent variable in place of the time, t. Here, 3 is the position along the optic
axis in the stationary frame. The reference frame moves with the reference particle and is
allowed to rotate such that the momentary direction of the 3 component points in the
direction of motion of the momentum of the reference particle, p0. The last variable has
been replaced by (Sr, which is the fractional difference in kinetic energy relative to the
reference particle and is a unitless quantity. At times it is necessary to describe the
motion of particles that have mass or charge that varies from that of the reference
particle. It is convenient to use the two variables,

r.,=5m=(m-mO)/mo r8=54=(q—qo)/qo,
to describe the fractional mass and charge difference, respectively. In the context of
beam optics, the system of coordinate system described above is sometimes referred to as

particle optical coordinates and we shall use them in chapters to follow.

13

1.4.3 The transfer map and characteristic symmetries

The method of calculating the final coordinates of a particle that propagates through
some region of space with a function acting on the initial coordinates is sometimes called
the mapping method. When working within the concept of maps it is more natural to use

the position of the reference particle along the optic axis, s, as the independent variable

instead of time. The method is based on the concept of finding some function, M(so,s),

that will act on the components of the initial position vector of some particle, ro= r(so), to
yield the vector components of the particle at a later point along 3. We can express this

action on the vector as,

r(s) = M(so,s)(r(so)), (1.7)

This function is sometimes called the transfer function, or transfer map, and its action on
the initial position vector is governed by the forces that the particle experiences relative
to the reference particle while being transported from so to s. A map can also act on
another to yield the overall map of back-to-back transport systems. If the particle
encounters two regions whose map is known for so to s; and s1 to Sf, then the map of the
system from so to Sf is give by

M(so,s,)=M(s,,s,)oM(so,s,). (1.8)
As long as the map is a continuous function along the region of s that is of interest, we
can expand the action of the map as a power series about the reference particle position.

This allows us to express the component, i, of the final position vector as,

8 1 8 l 8
fit =Zr0k{(ri’rk)+Ezr0.l{(ri’rkrl)+§Zr0.m{(’i’rk’lrm)+'"l}}r (L93)
k=l l=l m=l
where the values of the map coefficients (rork), (roan), (r,-,rmr,,,), are generally

functions of the independent variable s, but are evaluated at S] here. The coefficients are

14

related to the initial and final coordinates by partial differentiation as in a Taylor

expansion where

or“ 8er
(n,r,-)= a_ (r,.r,.r,,)= a—T— (1.9b)
70,,- r0.j :0 rod. r01 r0.j = roe =0

Since the system refers to an optical system, we can regard the linear terms (r,-,rk) as the
first order effect. All higher order effects are aberrations, and we refer to (man) terms as
second order aberrations, (r,-,rmr,,,) as third order aberrations, and so on. The terms in Eq.

(1.9a) may collected in groups that share the same order to obtain the form,

8 8 8
r.=Zror(r..r.)+— iZZrOJrO,(t;,r,r,)+%zzzr0mrwru(i, "$100+“

k=l l=l k=l m=l l=l k=l

Each term is may be referred to as a sum of monomial terms with aberrations of the same
order. In a realistic computation, expansion must be truncated at some order, NM, and all

higher terms neglected. Eq. (1.9) may then be expressed in the more simplified form,

_. n(l)

i(2) (3)

+r, +---+r,.‘”"’. (1.10)
Depending on the complexity of the fields that lead to the solution of the map, the
accuracy of the solution improves as the order, Nm, of the expansion is increased. The
subscript notation after the equal sign will hereby be neglected any time N,,,=l, except
when necessary.

In applying the map formalism one must be careful about the range in phase space in
for which the series converges. This is particularly important for higher N", values where
the forces need to be evaluated with higher precision.

When one considers only the linear terms of the expansion, the map may be

represented in the form of a linear matrix. For now, we will consider only the first six

15

components of the position vector. In such a case, the action of the map on the vector can

the be written as the matrix equation,

F -

 

 

=Ar0:

P(x.x)
(0.x)
(y.x)
(b.x)
(1.x)

 

_(5.x)

(x.a)
(a,a)
(y.a)
(b.a)
(La)
(5.0)

(x. y)
(a. y)
(y, y)
(b.y)
(l. y)
(5. y)

(Jab)
(a.b)
(Lb)
(b.b)
(Lb)
(rib)

(N)
(0.1)
(L!)
(M)
(1.1)
(5.1)

(1nd)"

(0.5)
(L5)
(b.5)
(1.5)

 

(6.6>__

 

, (1.11)

where the matrix A represents the map. The subscript for variable having the fractional

kinetic energy has been left out, so that 5 must be understood to represent (5(- If all three

variables had been included such that there is an 8 dimensional position vector, then A

would simply be an 8x8 matrix.

For any system made up of one or more optical elements, the first order map is of

primary importance since it is used to identify the most critical optical qualities of the

system. The matrix also reflects upon many of the symmetries of the motion. The most

notable of them is referred to as horizontal midplane symmetry and it is a result of having

only electromagnetic fields that are symmetric about the y=0 plane. As a consequence,

all cross terms between the horizontal and vertical plane will vanish such that the matrix

in Eq. (1.11) becomes [Berz85]

"(x, x) (x,a)
(a,x) (a,a)

A = 0 0
0 0
(1.1:) (La)

 

_(5,x) (a,a)

0 0
0 0
(My) (y.b)
(b.y) (12.12)
0 0
0 0

16

(L0 (15.5)-
(a,l) (a,5)
0 0
0 0
(1.1) (1.5)
(5d) (5,5)

 

.J

(1.12)

Maps of devices, such as quadrupoles, higher order multipoles, and electromagnetic
sectors where the motion of the reference particle is restricted to the y=0 plane, all fall
into this category.

If there is no bending of the reference trajectory at all, ie. no dipole fields, then a
double midplane symmetry is obtained. The result is that the terms located at the bottom-
left and top-right 4x4 sub-matrices of Eq. (1.12) vanish as well. Since there is then no
cross term relation between the x—a, y—b and l—J phase space planes, then three

remaining subspaces,

(x.x) (x.a) (y.y) (y.b) (1.1) (1.5)
A = , A = , d A = 1.13
’ [(a,x) (a,a)] ’ [am (12.12)] a“ ’ [(6.1) (5,5)] ( )

may be treated separately.

Another common symmetry that occurs in optical systems is that of rotational, or
axial, symmetry. Unlike double midplane symmetry in which the two planes of
symmetry, x—z and y—z, are offset by an angle, ¢=90°, rotation about the z-axis, axially
symmetric systems have symmetry about any arbitrary angle. In this case, all the same
terms vanish as in the double midplane symmetry, but additionally, all corresponding
terms in the x—a and y—b subspaces will be equivalent.

Maps of devices having no time-varying fields will have terms that in turn have no
explicit dependence on time. Hence, any terms in Eq. (1.9) with partial differentiation
with respect to the time variable, I, will, therefore, vanish so that (l,---) =0 except for
(U)-

Firrally, there is the condition in which either there is no change in the energy of all
particles in the beam or there is and equivalent change for all particles. This assumes that

all particles have the same charge as the reference particle, (qoe). An example of such a

17

system would be a DC accelerating column in which a potential drop between the

entrance and exit changes the energy of every particle by AK=-qoe(Vr Vo). For this case,

any terms in Eq. (1.9) with partial differentiation with respect to the energy variable, d

will vanish such that (5,...) = 0 except for (do).

If midplane symmetry about y=0 is imposed, along with explicit time-independence

and energy constancy, then the resulting form of the transfer matrix is

—(x, x) (x, a) 0 O
(a, x) (a, a) 0 O

0 0 (Ly) (rub)

0 O (b, y) (b,b)
(l, x) (l, a) O 0

0 0 O O

 

b

0 (L5)-

0 (0.5)

0 O

O 0
(1.1) (1.5)

0 (5,5),

 

(1.14)

Since all six variables are canonical conjugates, then the symplectic condition applies

as pointed out in Appendix A If we apply the symplectic condition to Eq. (1.14) we

obtain the following relations between the first order map elements:

(x,x)(a.a) — (x,a)(a,x) = Po,o / pOJ
(yr y)(b9b) _ (YobXb. Y) = [’00 IPOJ
(Ll) =,uo.1 ”10,0

(5,5) = Kw/Ko‘,

(x.X)(l.a) - (x.a)(l.X) = po—I‘gflfliufi)

0.0

(a,x)(l,a) — (a,a)(l,x) = -p°kL”°£(a,6)

0.0

18

(1.15a)

(1.15b)

(1.15e)

(1.15d)

(1.15e)

(1.151)

(a,x)(x,5) - (x.x)(a,§) __. :0»

 

(Lx) (1.15g)
0.!

(a.a)(x.5) - (x.a)(a.5) = 1;”

 

(l,a) (1.15h)
0.!

Here, poo, Koo, and poo are, respectively, the momentum, kinetic energy, and time-factor
of the reference particle at the initial position, while the subscript f denotes the final

position. It is assumed that the change in energy, (K0. I —K0'o), is the same for any

particle that is transported by the system. This can happen when all the particles are
accelerated over some common change in electrostatic potential. Since the fields are
static in time, we refer to this as DC acceleration. Here, a deceleration is understood in
the sense that the acceleration is negative, hence any change in energy is herby called
acceleration regardless of its sign.

When there is no change in energy, Eqs. (1.15a) through (1.15d) will all go to unity.
We note that this is a direct consequence of the invariance of phase space volume under
canonical transformations, as stated by Liouville's theorem [Landau76b]. It implies that
as long as the total energy of the system is a constant of the motion, then the action of the

map is effectively a canonical transformation on the ensemble of particles.

1.4.4 Focusing properties as determined from the first order map elements

Special focusing conditions are identifiable from vanishing terms in the map.
Furthermore, other map elements may become meaningful when one or more particular
elements vanish. Some examples are introduced now for the case in which the system
has particles that all have the same mass, charge and energy, and the electromagnetic

fields are time-invariant and symmetric at least about the y=0 plane.

19

 

Consider the path of the trajectories as seen from a side profile of the beam along the
x—z plane. We are interested in the values of the first order subspace matrix A, of a
system with special optical qualities. For example, if the rays enter the system parallel to
each other and exit such that all rays cross at the same x—position then we have what is
called a parallel-to-point condition, and the (x,x) term will vanish. The diagram in Fig.
1.3(a) illustrates a schematic representation of this condition. This condition indicates
that the exit plane, z=zf, is a focal plane. If instead the focal plane is at 2:20, then we must
have that (a,a)=0, and there exist a point-to-parallel focus as shown by Fig. 1.3(b).

Vanishing of the two off-diagonal terms in the x—a phase space are also of special
importance. The vanishing of the (x,a) term signifies that there is a point-to-point
focusing condition as depicted in Fig. 1.3(c). This gives an object-image relation
between the entrance and exit plane of the transport system. Any imaging device is
characterized by a spatial magnification factor, M,, which in this case has a value that is
equivalent to the term (x,x). The term (a,x) vanishes whenever the rays undergo a
parallel-to-parallel focus, as shown in Figure 1.3(d). In this case, there is an angular
magnification, N,, in which the ratio of the exit to entrance angle of the rays is given by
the term (a,a). An optical system of this type is considered a telescope focused at
infinity.

According to Eq. (1.15a), the determinant of A, must be equivalent to the ratio of the
incoming to outgoing momentum of the reference particle. This means that only the first
or second term of the equation is allowed to vanish simultaneously and restricts the
combinations allowed between the four focusing conditions already discussed. There are

only two combinations allowed. The combination in which (a,x)=(x,a)=0 is recognized

20

 

 

(C) (x,a)=0

 

 

 

 

(d) (a,x)=0

 

Figure 1.3 Schematic representation of several focusing conditions in which
elements of the map may vanish. (a) Represents parallel-to-point focus in which
(x,x)=0. (b) Represents point-to-parallel focus in which (a,a)=0. (c) Represents
point-to-point focus in which (x,a)=0. (d) Represents parallel—to-parallel focus in
which (a,x)=0.

as one with telescopic focus and having both spatial and angular magnifications, MJ, and
21

Nx, respectively. The (x,x)=(a,a)=0 combination simply implies that the z=Zo and z=zf are
both focal planes. The complexity that would arise from not having at least midplane
symmetry is the terms (x,y), (x,b), (a,y), and (a,b) would also be required to vanish to
obtain the focus conditions described above. Otherwise, these same arguments apply to
elements of the Ay subspace matrix if considering a beam profile along the y—z plane.

The (no) term characterizes the separation along the horizontal plane of particles that
differ in (2 whether its 5r. 6", or 5,. Devices that impose a bend along the horizontal
plane can cause the values of (x,o) and (a,o) to deviate from zero. If the system is
imaging, then we refer to the (no) term as the dispersion. More discussion on the
meaning of these terms will be provided when sections related to sector-field
spectrometers are encountered.

It should also be noted that the above conditions could also be extended to higher
order aberrations by considering the higher order derivatives of Eq. (1.9). This
discussion is left to other sections where the concept of the phase space ellipse is

introduced.

1.5 Numerical approach to computing rays

Some of the beam optical systems to be presented later have been studied with by
either ray trace computation or mapping. In some cases a comparison is made between
the results found by using both methods, thus a brief description of the numerical

approach used by each is suitable.

1.5. 1 Ray trace method of computation
In the ray trace method the trajectory for every particle must be computed by

numerical integration of the equations of motion. This is usually carried out in a

22

Cartesian flame of reference with the set of six ordinary differential equations (ODEs)

obtained from the components of [Reiser94a]

dR
—=v 1.16
dt ( a)

and

£_F —v(v-F)/c2 _qui+va—v(v-E)/c2)

. (1.16b)
dt 2m 2m

 

 

These are simply a rewritten form of Eq. (1.2) in which the electromagnetic force, F, is
replaced with Eq. (1.1). Notice that the last term in the numerator becomes negligible
compared to the terms in F for the non-relativistic limit, 7 = (1— v2 lcz)‘”2 z 1.

The commercially available code system, SIMION 7, was utilized in some of the
calculations. It has the ability to determine electric and magnetic potentials along a 3D
grid representing the region of space [Simion7]. The user enters the electrode and pole
boundary conditions, which the code uses to apply a finite difference technique to solve
Laplace's equation, V2<I>=O. Since there are no expansions involved in describing the
fields or trajectories, these computations are not truncated to a specific order as in the
map-based approach. SIMION solves trajectories by integration of the non-relativistic

form of Eq. (1.16) by iteratively computing
AR =1" vdt (1.17a)
and

Av=—l-I’th. (1.17b)
m o

23

It uses a fourth order Runge-Kutta numerical integrator with adaptive control over its
own progress. Adaptive control allows frequent changes in the integration step to
minimize computational effort while keeping some predetermined accuracy [Press92].

One advantage of solving by ray tracing is that it is straightforward to apply the
electromagnetic fields directly to the equations of motion without the need for
expansions. This is especially advantageous when applying finite difference algorithms
to determine the fields, which has made it possible to extend the technique for effectively
solving Vlasov equations [Birdsall85]. The fields can also vary with time as the
trajectories are being computed. Overall, this technique allows more flexibility in
treating the properties of the fields. There are set backs to using this method, however.

The most notable difficulty with ray tracing is the large amount of computation that is
necessary when there is a large number of elements over a long path and large number of
ensembles. Sector magnets, which are more suited for computer-based optimization, can
be physically large and more tedious to handle with ray tracing. Ensembles must be
limited in size, which makes it difficult to obtain accuracy. On the other hand, with
mapping once the elements of the map are determined, calculating final positions of rays
is simply a matter of evaluating polynomials. One major advantage lies in the fact that
the lower order coefficients tend to be of primary importance. Thus, it is advantageous to
Optimize an optical problem by successive evaluations and including more accuracy each
time by increasing, as well as increasing the level of electromagnetic field complexity if

necessary.

24

1.5.2 Computation of maps
There have been a number of computer codes with algorithms for determining the

coefficients of the Taylor expanded form of the map. A well known example is the beam
optics code TRANSPORT, which started off as being a second order code [Brown64] and
was later extended to third order [Carey92]. While TRANSPORT was evolving to third
order capability other codes, such as TRIO [Matsuo76] and GIOS [Wollnik87], also
become available with third-order capabilities. All these codes, however, relied on the
development of an analytical representation of formulas for the coefficients and were
based on a select list of elements with known field characteristics. With the advent of
specially made custom formula manipulators, the code COSY 5.0 was even able to push
the envelope even further by evaluating up to fifth-order terms [Berz87a]. These
methods, however, become prohibitively complicated at higher orders.

The advent of differential algebraic (DA) techniques has made it possible to extend
algorithms to higher orders and to more types of elements. The DA approach stems from
the transformation of crucial function space operations of addition, multiplication, and
differentiation, to a suitable set of equivalence classes [Berz90b][Berz99a]. The
implementation of these operations into the optics code, COSY INFINITY, allows
calculation of maps to arbitrary order [Berz90a]. From here on we refer to this code
simply as COSY, since it is used extensively in these studies.

To gain understanding of how COSY goes about evaluating elements of the map, we
should revert back to Eq. (1.9). Taking the derivative with respect to the independent
variable, s, to obtain component i of a set of expanded first order differential equations

(ODEs) given by,

25

dl' 8 1 8 8
l _ _ t 1
E‘fi"2’i (ri’rk) +0: (’i’rk )'+" Z ZrHZIrHlloi’rklerkH]) +"'
k=l k[2]=lk[l}=l

1 8 8 . (1.18)
+7 2 m2’11le"'rkllloivrklel"'rklll).
,,.th,,1=| klll=|

If the equation is written in unexpanded form then we obtain that
Jr.-
E:fi(rl,r2,r3,r4,r5,r6,r7,r8,3) =fi(x,a9y9bslyaxsam9§qts) 9 (1°19)

where the coordinates of the function f may depend on s. We restrict ourselves to the

case in which 5,. and 5, are independent of s, i.e. 5,,I '= 5q '= 0. We want to show that the

set of ODEs implied by Eq. (1.19) are related to the time dependent Hamiltonian, which
are in turn obtained from the equations of motion.

To make this relation realizable we must carry out a Legendre of the type discussed in
Appendix A to switch the independent variable from s to t. In fact, the transformation
would essentially be the reverse of the one in the appendix in which the equations were
transformed from t to s dependence. This type of transformation should yield a new set

of ODEs,

dr.
j=hi(x,px,y,py,s,ps,t). (1.20)

where the coordinates are also regarded as functions of time. We can compare the h;

components with the system of Hamiltonian equations,

ban/an, p,=—aH/ax
i=aH/ap, p, =-aH/ay. (1.21)
s' =aH/ap, p, =—aH/as

where the dot notation stands for d/dt and the former independent variable, s, and its

corresponding momentum, p,, have now been transformed into variables. Although we

26

worked about it backwards, it has been shown that the Hamiltonian of the system can be
transformed to give the ODEs of Eq. (1.19), which in turn may be expanded to the form
of Eq. (1.18). Integrating each term in Eq. (1.18) yields the corresponding terms in Eq.
(1.9), and in effect, solves the expanded form of the equations of motion in map form.

The form of the Hamiltonian in (x,a, y,b,l,5x,s) space and the corresponding
equations of motion have been worked out by Makino and Berz in full 3D curvilinear
coordinates [Kyoko98][Berz99b]. The equations are more involved in full curvilinear
form, hence, we do not elaborate on them here. For these studies, we restrict the
discussion to the form of the equations of motion under midplane symmetry conditions
about the y=0 plane. The motion of the reference particle is restricted to this plane, and
the instantaneous curvature of the motion is given by, h(s)=l/p(s), where p is the radius

of curvature. Under these condition the equations of motion are given by

x'= a(l+h.x)p0/pz (1.22a)

y'= b(1+ Ivor0 / p, (1.22b)

1': (1+5m)(1+1u)1i+—”—& (1.22c)
o Pz

 

B
a'=[(1+6m)1+" p° E’ — ’ +b&—£-J(l+hx)(l+o‘q)+h£i (1.22d)
1+7lo P; 2'50 IMO Pz Zuo Po

 

E
b'=[(1+6m)1+” p0 y + 3’ -a&i](l+hx)(l+5q) (1.226)
1+7lo Pz 2'50 IMO Pz IMO

5,; :0. (1.221)

27

where the prime still represents the derivative with respect to s and all parameters with

subscript 0 are of the reference particle. The quantity 7] is a measure of relativity as given

 

by,
77= K2 ___ Ko(1+5x)2+AK(x,y,s,t). (1.23)
mc moc (Ii-5,”)
The quantities,
150 =M and IMO =& (1-24)
40" 409

are the electric and magnetic rigidity, respectively, of the reference particle. We can

evaluate the momentum ratio that appears in each of Eqs. (1.22) by

n=fi,,5,21<2_+a_,2_,2, (1.25)

 

. "' mam.)
Equation (1.221) vanishes since we have assumed that there are no fields with explicit
time dependence in the system. This does not necessary mean that the kinetic energy of a
particle cannot vary along the motion. Indeed, if a particle enters a region in which the
electric potential varies in space (not time), then it will experience a change in kinetic
energy, AK. This is reflected by the form of 7) in (1.23) where AK. Suppose we assume
initially that there is explicit time dependence, then the change in kinetic energy will vary

33,
AK(R(t),t)=q f E(R(t),z')- «7) dr. (1.26)

If we turn off the time dependence, then E = —VV and AK can be evaluated from

AK(x,y,s) = —qe(l+5q)V(x,y,s). (1.27)

28

This assumes that the potential has been expressed in the curvilinear coordinates and that
the potential is specified relative to the entrance position such that V(x,y,0)=0.

At this point we can compare Eqs. (1.22) and (1.18) and observe that by expanding
the components of the electric and magnetic fields on the right side of each equation in
(1.22), one can form a relation between the s derivatives of the map elements and the
equations of motion. The code COSY uses a numerical ODE integrator and DA
functions to determine each element of the map as described by Berz and Wollnik
[Berz87b]. At its core this ODE integrator utilizes an eighth-order Runge-Kutta
algorithm with adaptive control over its own progress, making frequent changes in its
step size, similar to the fourth order integrator of SIMION.

The code has integrated features that allow it to algebraically determine elements of
the map without numerical integration. This can substantially reduce computational time;
especially in the first stages of a problem where not a lot of accuracy is needed. The
method is based on an analytical solution in which it is assumed that the particle
experiences no forces along its direction of the velocity and that the curvature remains
constant along the motion. This is true of a number of situations; the most obvious one
being that of a drift along a field free region. Another is the case in which fringing fields
at the boundaries of a homogeneous magnet are ignored. In this case the magnetic field
is always parallel to the y—axis and the field changes abruptly at the dipole boundaries.
This is very similar to the case of a light ray entering a glass medium with constant index
of refraction, except there the curvature is h=0 and the velocity changes abruptly at the

boundary. Indeed, COSY also has features that calculate the maps of glass optical

29

 

systems [Berz99c]. For both glass optics and magnetic dipoles the exact shape of the
boundary must be specified.

In any electromagnetic device there is always some level of fringing, and in fact, may
depend exclusively on fringing for its focusing strength. Any optical elements with fields
varying along the optic axis require numerical integration of the ODEs along the s-axis .

Devices such as dipoles and multipoles tend to have extensive regions along the optic
axis over which there is little variation along the s—axis. This region depends on the size
of the gap and any shunts, or clamps, that may be applied in the region just outside of the
pole region [Wollnik87a][Hiibner70]. The regions in which fiinging is negligible are
calculated using the efficient DA methods and the fringing regions may be treated in a
number of ways. For computational efficiency one may use a symplectic scaling option,
and for accuracy and flexibility one may specify the coefficients of an Enge function to
describe the field dependence on s [Hoffstatter96]. The equation for this function is
given by,

I
1+exp(a, +a,(s/G)+--~+a6(s/G)°)’

 

where G is the gap width of the dipole, or diameter if it is a multipole. The constants a)
thru as are the Enge coefficients. We use the available default values that derive from
measurements taken of a family of unclarnped multipoles at the PEP facility [Brown8l]
and plot the corresponding Enge function in Figure 1.4.

Before going on to the next section, a few things should be mentioned about the other
possible beam interactions. The effects of energy loss by emission of synchrotron
radiation are not accounted for in the equations of motion in COSY, nor are the self-

interaction forces due to space charge. These effects will not be important for our

30

 

 

 

 

 

 

1,0 ‘ —dipole 4
1 “t‘ """" quadmmle * Figure 1.4 Enge
03f i, 4 functions for dipole and
i ‘ quadrupole elements as a
0.6- function of position over
3 ' gap width, G. The Enge
‘K 0.4- coefficients used were
‘ derived from
0.2- measurements as cited in
i the text.
0.0 -
-a 32 -'1 6 i 2 3

applications, but can in principle be formulated into the equations [VorobievOl]
[Berz90a]. COSY currently accounts for the forces from electromagnetic field
interaction with a particle's spin, but is a negligible effect here; therefore, it is left as a
reference [Berz99d][Makino99].

Generally, the COSY code system offers a variety of options that makes it
advantageous to use over other applications with maps. For one, the user interface is
written as a high level language to let the user write custom programs. One can readily
create custom optical elements in this way. There are also a variety of algorithms for
optimization of optical system. They allow one to specify any objective function that is

to be minimized [Berz97].

1.6 The phase space of beams

If we neglect the rotational degrees of freedom, then the motion of any particle may
be specified by its vector position and momentum. In the case where there exists a large
group of particles it is useful to apply the concept of phase space volume. In this

macroscopic description one specifies the phase space volume that an ensemble of

31

particles occupies. The phase space can be fully described by some six-dimensional
volume with the coordinate variables X, Px, Y, Py, Z, and P2. It is useful to apply other
coordinate systems from beams, and it has already been demonstrated that by a canonical
transformation the system may also be described by the variables x, a, y, b, l, and 5:.
From now on, these variables will be used to describe phase space, unless otherwise
specified.

It is customary to describe the phase space by the amount of area that the beam
occupies in each of the three subspaces, x—a, y—b, and l—&. The first of these two
subspaces are said to lie along the transverse planes since the direction of motion in this
plane is perpendicular to that of the reference particle’s. The units are. usually in m-rad,
or mm-mr. The direction lying tangent to the orbit of the reference particle is referred to
the longitudinal direction and the phase space area along the l—& plane is usually
referred to as the longitudinal phase space. The units are given in meters with an
understood fractional quantity. To obtain units of energy and time one needs to multiply
quantity by the energy of the reference particle, Ko, and divide by the factor, #0. The

units commonly used are keV for energy and us for time.

1.6.] The concept of the phase space ellipse
Since beams tend to exhibit bell shaped distributions along any of the phase space

variables, contour plots along any subspace will generally exhibit a family of ellipse-
shaped boundaries. The contour lines indicate the level of beam intensity and the
integrated intensity as one goes to a lower and lower intensity level is sometimes used as
a figure of merit in beam applications. Further discussion about this topic is left to

another section below.

32

We restrict the following discussion to the x—a subspace noting that it also applies to
the other two subspaces in the same manner as treated here. The equation of the ellipse

representing the phase space boundary may be expressed by the equation,

7xx2 + 2axxa +flxa2 = 8x , (1.29a)
orinmatrix form,
a
[x a]- 7‘ x - x =(r)T-o-r=e,. (1.29b)
0'. l3, 0

Here, yx, ax, and ,6; are known as the Twiss, or Courant-Snyder, parameters and in order

to have only one unique solution for some given 8, we demand that the determinant of the

sigma matrix, a, be unity; i.e.

my, —a,2 =1. (1.30)
The area bounded by the ellipse will be referred to here as the emittance area and is
given by 1%}, where 8, will simply be called the emittance. One should be aware that
other references might refer to the area as the emittance.

It is useful to solve for a and x from (1.29a) to obtain the two quadratic solutions,

_-a,xi‘lflxex-x2 _—a,ai‘l7,e,—a2 (131)

a— and x—

flx 7‘

The factors inside the square root must remain positive to be within the region bounded
by the points of extremum at x,,. and a,,, as illustrated by the Fig. 1.5. The extrema are

solved for by setting the quadratic terms to zero and applying condition (1.30) to obtain,
x", = ,8, and a = 7,81 . (1.32)

By inserting each of these into equation (1.2%) we obtain the other two terms shown in

the same figure,

33

ae=—a',r Exlflx and x,=—a, 83/7,. (1.33)
Hence, the width along the x— and a—axes of the ellipse are determined by the
parameters ,6, and x, respectively. The parameter a, quantifies the tilt of the ellipse in

the sense that the ellipse in Figure 1.5 has a value a,<0.

1.6.2 The normalized emittance

Take the situation where we have an ensemble particles forming an upright ellipse at
s=so in the x—a phase space with emittance, Em = xm'oam'0 , and reference particle
momentum, poo. The ensemble is transported through a telescopic focusing system that

applies DC acceleration to obtain the final momentum, Poi. as it exits at s=sf. As long as

there exists at least midplane symmetry, then equation (1.15a) applies and we must have
that (x, x)(a,a) = p0.0 / poJ . Since (x,x) and (a,a) are the magnification along the x— and
a—axes, respectively, we must have that

Po.o

P0,}

811.; = meamJ = (x,x)xm‘0 - (a,a)am.0 = 83.0 (1.35)

This result implies that the emittance changes by a factor proportional to the ratio of the

initial to final momentum. If there is a positive acceleration then the emittance will be

 

 

 

a Figure 1.5 Diagram illustrating
(xe’am) the points of extremum in the
\ phase space ellipse along the x—
(xmta) a plane. The tilt is such that
a,<0.
. x

 

 

34

reduced by this factor, and vice-a-versa for negative acceleration, or deceleration. Note
that we assumed a telescopic system with upright ellipses for simplicity. This result,
however, applies to any system of time-independent fields.

There is an invariant quantity that becomes apparent if both sides of Eq. (1.35) are
multiplied by the final momentum. We define this invariant quantity, gun as the
normalized emittance such that,

8mmc = 8L, po'f = xmoam‘0 , (1.36)

where the constant mc may be divided out to obtain the same units as the standard

emittance. Since there is invariance along 3, then the subscript for position may be
dropped and we can simply write the normalized emittance as
em =exyofio. (1.37)

The normalized emittance is essentially the phase space occupied by the beam in the rest
frame of the reference particle. As long as there are only conservative forces acting on
the system of particles and there is no self-interaction or interaction with any other
external medium, then this quantity will remain constant. Non-conservative interactions
can be caused by a number of things. For example, there are collisions with walls, foils,
or residual gas particles that generally tend to cause an increase in the normalized
emittance. The emanation of synchrotron radiation will generally cause a decrease,
although negligible in most cases. It is most prominent in the case of highly relativistic
light particles, such as electrons. Finally, external forces that originate from time-varying

potentials will generally cause some increase in the normalized emittance.

35

1.6.3 The transformation of the sigma matrix
The concept of the phase space ellipse is used to describe the effects caused by of

optical elements on the beam's phase space. It is mainly restricted to first order
transformations since higher order effects will distort the boundary to shapes that deviate
from the equation of an ellipse. A number of examples of this effect will be given later,
but for now we accept the restriction in order to discuss the effect on the sigma matrix
under such linear transformations.
As follows from (1.29b), the equation of the ellipse for any subspace at some initial

s=so can be written in terms of its respective sigma matrix, do, by the equation

(r0)T cor, =e0. (1.38)
We suppose that in going from so to Sf the particles in the beam experiences DC
acceleration and that map of the system is given by A such that,

Aro =rf, (1.3%)
and its inverse defined in the sense that,

A"rf =ro. (1.39b)
If we assume midplane symmetry and restrict our attention to only the x—a subspace,

then as suggested fi'om (1.13),

A = [(x’x) (“0] (1.40)

(a,x) (a.a)
We should like to determine the inverse of A such that A"A=l, where I is the unity

matrix. First, note that

|: (a,a) -(x,a)]'|:(X.x) (x90):l_ p0,0 I (141)

—(a, x) (x,x) (a,x) (a,a) — PM

 

36

where the ratio of the initial and final momentum outside of the unity matrix results from

Eq. (1.15a). The matrix on the left is similar to the inverse matrix if we divide it by this

factor to obtain,

trl =p_°-f_[ (“"0 "(x’“)] , (1.42)

p0.0 — (a,x) (x9 x)

The equation of the ellipse at Sf may be expressed by,

p
(r,)To,r, =2, =—°'9-£0. (1.43)

where the results from Eq. (1.35) has been applied. If we use (1.3%) to express Eq.
(1.38) as,

(rf)T((A")TeoA" , :30, (1.44)
then a direct comparison with (1.43) implies that sigma matrix at Sf may be evaluated

from the initial sigma matrix through the transformation,

a, =ip°'—°(A‘l )TooA'K (1.45)
P0,]

Writing this equation out explicitly for the x—a subspace, we obtain that

[7x,f axJ]:

a

"f '6” . (1.46)
men—fl (a,a) -(x.a)]_[7..o 01.0].100.) [(x.x) 0.0)]

p0,; Po,o —(a,x) (x,x) “11.0 firm pop (a,x) (a,a)

 

Carrying out the matrix multiplication to obtain the elements of the final sigma matrix in

terms of the elements of the map, we obtain the following set of equations:

7,. f = mine (cw)2 - 261'...0 (0.x)(a. a) + ,6”, (a. x)2] (1.47a)

0,0

37

P0,]

0.0

l- 7..., (x.a)(a. 0) + are ((x. x)(a.a) + (x.a)(a.x))- 13,0 (x, x)(a,x)] (1.47b)

 

a", =

P0. f

0.0

,6” = [73.0 (x, a)2 — 20'”, (x, a)(x, x) + ,6”, (x,x)2] (1 .47c)

 

The simplest example of an evolving ellipse under transformations is that of a system
where there are no fields, i.e. a drift. If the reference particle undergoes a shift along the

optic axis of length, Lo, then the map is simply given by,

_ 1 Lo
Aug—[0 I]. (1.48)

The spatial envelope within which the beam is bounded is given by x,,,. As a function of

the drift length, and according to Eq. (1.32) it must take the form

 

xm (L0) = Jew (7m (lolz —2ax,0(l‘0)+flx.0)' (1°49)

Also, inserting the matrix values into Eq. (1.47a) will show that the term, 7,“; , remains
constant. An example of this is depicted in Fig. 1.6 where an initially upright ellipse
propagates through two drifts of the same length. Notice that the ellipse expands along
the x but remains constant along a.

To illustrate the effects that focusing and defocusing elements have on the phase
space ellipse a rather complex system of electric quadrupoles is depicted in Fig. 1.7. The

optical system extends over a total length, L, and has mirror symmetry about the center

f a Figure 1.6 The evolution of
the x—a phase ellipse through

two equivalent drifts.
-—.-
x 67 fl
1 2 3

38

position, s=U2. Each triplet is also symmetric about its center position, and the fiinge
field effect for each quad has been accounted for in calculating the maps with COSY.

The beam is assumed to be made of ions of 100 keV with charge state q=+l. The

 

   

I
l
l
!

 

 

 

l.“ ;A4‘SI
2| 31

Figure1 1. 7 Sextet system of total length L made up of electric quadrupoles as described in
the text. Plots of the ray trajectories along the x—s and y—s planes are shown in the top
two plots, while snap shots of the x—a phase space ellipse are shown for both a first order
(line) and a fifth order calculation (dots). Other setting are as follows: quad apertures
radius = 2.5 cm, quad length = 15 cm, L1 = 30 cm, L2 = 20 cm.

 

 

39

trajectory path of the ions along the x—s and y—s planes are shown in the top two plots
of the figure for quad strengths Q1=—0.74 kV and Q2=1.3 kV, where positive polarity
implies a focus along the x—plane. A fitting algorithm had been used to find Q1 and Q2
such that the sextet has telescopic focusing in both the x—a and y—b planes at the exit.
This can be seen by the way each ray coming in parallel at the entrance comes out
parallel again at the exit where (a,x)=0. Also, rays that are diverting at the entrance
converge at the exit, since (x,a) also vanishes there.

Only the phase space ellipse of the x—a plane has been illustrated in the bottom plot.
Snap shots of the x—a phase space ellipse are shown for both a first order (line) and a
fifth order calculation (dots) at positions labeled 1 through 7. At s=0 the particles all lie
along the boundary of the upright ellipse with xm=4 mm and am=2.5 mr. The higher
order aberrations cause projections of filaments to form as the particles diverge away
from the boundary of first order. This is seen more clearly at positions 4 and 7 in the
form of an asymmetric filamentation. This is caused by third and fifth order geometric
aberrations, such as (x,xxx), (x,aaa), (x,xxxxx), and so on, which as expected results from
elements focusing with electric fields [Wollnik87b]. The effects of the aberrations have
been exaggerated by a factor of 20 for illustration purposes.

Some last details in the first order ellipse that are important to notice are the focusing
and defocusing effects. The elongation of the ellipse in going from point 2 to point 3 is
caused by defocusing of the first quad. The second quad focuses in such a way that the
ellipse is "flipped" and the particles are converging afterward. The combination of
defocusing and focusing causes the ellipse to become upright at the midpoint of the

system (point 4). The final ellipses show some distortion at the extremes. At times

there is some confusion as to whether these types of geometric distortions also cause the
bounded phase space area to change. The answer is no, since according to Liouville's
theorem the volume of phase space remains constant even though the shape generally
will not [Reiser94ab]. Hence, the boundaries of both the first and fifth order calculations
remain constant since poo/poi: l, which is a consequence of Liouville's theorem that was
pointed out in Appendix A.
1.6.4 The rms emittace and higher order effects

Beam calculations based on phase space dynamics must be carried out using particle
ensembles of finite size. An ensemble of particles represents a sample taken from the
volume occupied in phase space. To analyze a sample it is necessary to use statistical
methods for evaluating the effects from fields. A simple statistical method for evaluating
first order properties of the ellipse fi'om rms values is introduced in this section. It is
based on the concept of evaluating the rms emittance, 2" . Here, it shall be applied only to

the x—a subspace, E", , but it is easily extendable to the y—b and l—5 subspaces.

The first step in evaluating the properties of the rms emittance is to calculate the

following rrns values given that there are N particles in the ensemble:

N
<x2 >='1172(xi—< x>)2 (1.50a)
i=1
2 1 N 2
<a >=7v—Z(a,—<a>) (1.50b)
I N
<xa>=FZ(xi-<x>)(ai—<a>) (1.50c)
i=1

41

The three quantities are considered to be the second moments, while <x> and <a> are the
average values, or the first moments along the x— and a—axes, respectively. The

corresponding rms emittance is related to the second moments by,
Ef=<x2><a2>—<xa>2. (1.51)

This quantity is invariant under linear transformations where there is no acceleration. In

cases where there is DC acceleration but the transformation is still linear, the final rms

emittance is proportional to the initial by the factor, p0.0 / p0,, . This is similar to the case

of the initial and final emittance as shown by ( 1.35). In fact, the emittance and the rms
emittance are linearly proportional to each other and remain so as long as there are only
linear transformations acting on the particles.

The constant of proportionality between Ex and 13‘,r depends on the type of density

distribution in phase space that is assumed. To demonstrate how the rms emittance varies
with the type of distribution three plots have been generated with differing density
distributions and illustrated in Fig. 1.8. Each distribution has an equivalent rms
emittance, and an ellipse whose emittance is equivalent to the value of the rms emittance
has been drawn for each of the three.

The leftmost distribution is called a boundary type since the particles are randomly
distributed along the boundary of an ellipse. It may be generated using the method of
spheres in 2 dimensions [Muller59]. The rrns emittance value turns out to be one half of

the emittance of the ellipse whose boundary is being populated. This is shown by the

ellipse, labeled E, = 22"

X,

in the first plot. A waterbag-type distribution, generated by

using the method of spheres [Lewis75], is shown at center and has only about 22% within

the ellipse of ms emittance, while the Gaussian-type, on the right, has 68%. Both

42

Distribution type:

 
  
  

Boundary
2. =3. (0%) e, = E. ‘22
e. = 25', (100%) E, = 4?, (95°W ' t

Figure 1.8 Plots of three differing phase space distributions with equivalent rms
emittance. The type of distribution is labeled above each plot and at the left side of
each plot is the percentage of particles lying within the corresponding ellipse of
enrittance ex.

waterbag and Gaussian distributions contain 95% of the particles within an ellipse of

The boundary-type distribution is useful for looking closely at higher order effects
since it samples the extremities of the phase space. Such use was demonstrated above
with the x—a phase space plots at the bottom of Fig. 1.7. The waterbag-type distribution,
which is just an extension of the method of spheres to 6D, is an almost uniform
distribution. Although, it does tend to form a low-density pocket towards the center of
the ellipse, the ease in generating the waterbag distribution makes it an attractive option
for many applications. The Gaussian-type distribution is one that best describes many
experimentally observed beams and is ideal to use for more precise calculations. A
method prescribed by Box is used to generate the random distribution of this type
[Box58].

Once the rms emittance has been evaluated, the rms emittance may be divided out of

each of the values in (1.50) to evaluate the Twiss parameters by the following:

43

 

fl: = ~ (1.523)

7; = ~ (1.52b)

 

 

(1.52c)

Solving for <xz>, < a2) and < xa> from the above equations and inserting the results into
(1.51) proves that this results is true. The negative sign in (1.52c) keeps the tilt
orientation consistent with Fig. 1.5.

Something that that should be kept in mind when applying higher order effects is that
the rrns emittance may grow even though the actual emittance remains constant. This is
due to the fact that the rrns values are very sensitive to particles that may form outside the
first order ellipse boundary. This can become especially troublesome for repetitive
systems, such as storage rings, where very low-density halos may form around a high
density region. Despite this fact, the rms method remains an invaluable tool for
evaluating beam systems. It offers reliable first order knowledge of the ellipse

orientation, which is useful in matching one optical system to another.

Chapter 2

DESIGN OF AN ISOBAR SEPARATOR USING AXIALLY SYMMETRIC
ELEMENTS WITH ACCELERATION

In this chapter we describe the design of a high resolution mass separator that uses
accelerating axially symmetric elements so that the system is an achromat. The RIA
facility will need exceptionally high mass resolution to separate masses at the level of
isobar mass differences. A brief description of other types of mass separators will be
given along with reasons for choosing a spectrometer with a decelerating column, which
we call a dual-potential spectrometer. Much of the chapter will be devoted to details of

this spectrometer along with the factors that affect its performance.

2.1 Isobar-1c purity and mass separators

Much of the success of future rare isotope facilities depends not only on the intensity
of the isobar of interest, but also on the purity of the beam. Isobars have very small mass
differences, and beams coming from ISOL targets will be susceptible to cross
contamination by other ions of similar mass to charge ratio or ions of some energy that
may cause a contaminant to cross over into the region of others. Furthermore, for isobars
far from the line of stability there will be tails from other more intense ion species that
will also cross over. To make matters more complicated, the mass window will have to
be large enough such that transmission of the wanted ions species is maximized. Under
these stringent conditions one is many times forced to speak of mass separator devices in

terms of the level of purification from contaminants, as we shall do so in a later section.

45

We shall explore the feasibility of high mass resolution separators with high
transmission for the RIA facility. High transmission with a resolution of m/Am=20,000 is
the base line goal for the present study. This resolution corresponds to a mass excess
difference of 5 MeV at mass A: 100.

Other methods exist as possible alternatives to the scheme that will be presented in
the next section and a brief discussion should be devoted to thenr. For many chemical
elements for which the ionization potential is low enough, there is the possibility of using
laser resonance ionization techniques to obtain high chemical selectivity [Alkhazov89]
[Koester02]. This is an ideal way to suppress isobars that have differing chemical
properties from the species of interest. The efficiency is usually between 1 and 7% and is
mostly limited to elements with of ionization potentials <7 eV. Alternatively, using
surface ionization can be almost 100% efficient for ionization potentials of less than
~5eV. This method is limited to about one tenth of the entire spectrum of elements
[Hageb092]. A combination of these techniques and other more general ionization
techniques having less selective ionization have to be employed to extend the amount of
available beams. Consequently, it is helpful to shift the mass selectivity to devices lying
beyond the ionization region.

Magnetic sectors provide spatial separation between masses and have been the heart
of most isotope separators for several decades [Smythe34]. One limitation stems fiom
increasing higher order aberration effects in going to larger acceptance sector magnets.
Other limitations stem from the fact that all ion sources impose some level of energy
spread on the beam and also from instabilities of the power supplies. The effects, which

appear as smearing and chaotic shifts of the spectrum, will be magnified along with the

 

resolution. The most fundamental problems, however, are still the energy dispersion and
geometric aberrations. Reducing geometric aberrations will require strategic
superposition of multipole fields, and energy dispersion must be eliminated designing a
spectrometer that is achromatic.

Cyclotrons carry some degree of achromicity by using RF electric fields and have
been used for mass selection. By extending the number of turns before extraction, they
have been known to give a mass resolution of as high as ~104 [Huyse95] [Chartier97].
One of the drawbacks is that the transmission efficiencies turn out to be rather low (3 to
5%). Furthermore, the output energy range of cyclotrons is limited and some
experiments may require that ions arrive at the slowest velocities possible, such some in
nuclear astrophysics.

Another type of RF device that offers a similar type of mass dispersion but without
the longitudinal acceleration is the quadrupole mass separator (QMS). They are
especially appealing since they do not require magnetostatic fields, which allows over all
system dimensions relatively minute in comparison. Unfortunately, some of the best
QMS designs to date have been know to achieve resolutions of no better than about 300
under DC beam operation. Studies that focus on improving QMS performance seem to
indicate that the distorted fringing fields at from the ends of the rods introduce
aberrations that effectively increase both the transverse and longitudinal emittance of the
beam [Takebe95]. Such effects also cause the overall acceptance and transmission to
suffer. Some of these distortions may be eliminated by refining the rod design and
alignment; however, a recent study shows that the rod design requires at least a 1 pm

position accuracy to approach a mass resolution of ~2000 [Yoshinari95]. Despite such

47

improvements much work still remains to improve the acceptance, thus such devices
remain impractical for isobar separation.

Considering such technical dilemmas with the other types of devices, the use of
single-pass magnetic bends remains the best alternative. By coupling magnetic dipole
systems with DC electric field devices, it is possible to compensate for energy spread
effects. There will still exist a multitude of technical barriers to eliminate in going with
magnetic sectors. For example, the uniformity of the fields has to be maintained to a high
level. We shall explore this problem with the use of a Poisson solver to determine the
fields that should be expected from some of the sector designs. Issues concerning control
and stability must also be addressed, since the object and image sizes are necessarily
small.

The design goal of the isobar separator described below are a mass resolving power
of at least 20,000 given a transverse emittance of 101: mm-mr for ions at 100 keV/250
amu (1 mm entrance slit width and 1:20 mr maximum divergence) and a :10 eV energy

spread.

2.2 Achromatic mass separators

As mentioned in the previous section, when considering mass separators at the level
of isobar mass differences it is necessary to include the effect of the beam energy spread
from the ion source. It is possible to obtain some energy spread due to the ripple of the
ion source power supplies; however, the voltage ripple from modern power supplies can
be suppressed to about the 105 level. Instabilities of the source platform potential have
been known to be caused by the driver beam as it is injected in the form of intense pulses

into the target, but this should not be a problem with the RIA driver since it functions

48

essentially in continuous wave (cw) mode. The most probable cause of energy spread is
the thermal energy spread associated with plasma type ion sources. Many low charge
state plasma sources can impose energy spreads that are on the order of ~10 eV. Thus,
considering a spread in the energy at the 104 level will be necessary. Magnetic sectors
disperse the ions according to magnetic rigidity, [App/qe, thus the mass and energy
dispersion are equivalent; i.e.

(x. 5..)=(X. 5t) (2. 1).
Because of this additional dispersion in energy the resolution of the spectrometer will

suffer unless the (x, 5‘) can be eliminated.

The principle of an achromatic mass separator can be understood in terms of the first
order transport map in matrix form. Consider the position variable in the horizontal
plane, x and a, in a subspace that also includes the mass and energy variables, 5,, and do
respectively. If we consider the position vector (x,a,5.,&), then the transfer map can be

expressed as,

_ (x, x) (x, a) (x, 5,) (x, 5...) '
A: (a. x) (a. a) (a,ax) (a, 5...) 2 2
(610x) (6,90) (6K96K) (6,065") ( - )9

_(5,,,,x) (5m,a) (6,,,5,,) (6,,,5,,)j

 

 

which yields the final position vector when we take its product with the initial position
vector. This subspace is suflicient to consider as long as mid-plane symmetry is
preserved. Formally, an achromatic system should have no energy effects whatsoever;
however, here we are only considering a first order achromat in which (x,&) vanishes.
Higher order achromats require that higher order aberrations with t} dependence also

vanish. Also, note that if the term (a, 5‘) does not vanish, then particles gain angular

49

divergence hour the energy difierences. This is not acceptable if the particles are to
continue on through further beam system, such as the post accelerator. A fully

achromatic system must have both terms vanishing.

2. 2. I Double-focusing spectrometer
We look first at the properties of a double-focusing mass separator [Yavor97a] like

the one illustrated in Fig. 2.1. Note that there are other ways to arrange the magnetic and
electric fields to obtain the similar properties of this system [Nolen84]; however, we use
this particular one for its simplicity. With this system we can evaluate the map of the
magnetic and electric fields, A; and A5, respectively, in separate form. The product of
the two,

A =-' A5 ' A3, (2.3)
yields the map of the total system. Imposing point-to-point horizontal focusing at B and

C results in the energy dispersion term

(L5,) = (x.x)5(x.5x ), + (m5, )5 (2.4)-
Requiring that this term vanish implies that rays of difi‘erent energy will get refocused at

C as illustrated by the figure.
Note that for the system shown in the figure the term (a,&) does not vanish, and rays

of different energy are converging as they approach the focal plane and will diverge right

Figure 2.1 Double-focusing spectrometer
with rays of multiple divergence as well
as multiple energies. Rays are focused in
the horizontal plane but split by varying
energy at point B.

 

   

f

B-section E-section

f c

A

 

 

 

 

  
 

50

after. This is only a simple illustration of an achromat, but there actually exist solutions
in which systems with magnetic and electric sectors in tandem form a fully achromatic
mass separator. One such design has been previously introduced for the RIA system as a
possible candidate for an isobar separator [Davids94]. Studies of the performance of
previously constructed designs using large dipoles but at lower resolution has been
discussed elsewhere [Davids89] [Davids92].

Although such designs are conceptually sound, there are some technical difficulties
with very large electric sectors. Small mutual vertical inclinations along the surfaces of
any pair of electrodes tend to misalign the beam from the mid-plane and will give so
called “parallelogram-type” defocusing [Yavor97b]. This effect has been known to cause
severe losses in the resolving power [Matsuda77]. In principle, small corrections can be
made on the electrodes to obtain a more uniform field without misalignments; however,

there are inherent technical difficulties when working with electrodes at high potential.

2.2.2 Dual-potential spectrometer
To avoid the efiects of electrostatic condensers another method of energy focusing

was devised which uses at least two stages of magnetic separation at difl’erent potentials
[Ciavola97]. So-called dual-potential spectrometers have been used in the past for the
purpose of eliminating unwanted scattered particles, but they were not achromatic
[Wollnik95]. The present design is a dual—potential achromat with a layout illustrated in
Fig. 2.2.

The system is broken down into four sections. The beam enters section H (points A-
C) at KH=100 keV and thereafter is decelerated through an immersion lens system at

section I (from C to D). The third stage (points D-F) lies at a 90 kV potential on an

51

isolated platform so that the beam drops in energy to KL=10 keV. The potential may vary
as long as the ratio of the initial to final kinetic energies remains constant at KHIKL=10.
The radius of the magnetic sectors in section H (labeled DH) are RH=2.5 m, while those
of section L (labeled DL) are just R” NE. This scales with the rigidity to keep all the
pole tip field strengths at the same value. All sectors bend through an angle of 60°. The
DH sectors have positive entrance and exit edge angles of 24°. Those of the DL sectors
are slightly lower at 23° for both entrance and exit. Finally, the last section is simply the
reverse of section I and accelerates the beam back to K" as the beam exits the isolated

platform.

 

”T -. \ M011 1" [A-C]
M" / Sectionl [C-D]
Section L [D-F]

D - dipole
Q - quadrupole
M - multipole

 

 

- QI-Il

 

 

 

A_l

 

Figure 2.2. Layout of dual-potential spectrometer. The spectrometer is broken down
into 4 sections as described in the text [PortilloOla]. Overall footprint is 20m x 30 m.

52

Magnetic multipole fields are imposed at the midpoint of both sections H and L (MH
and ML3, respectively). The multipole MH provides quadrupole, hexapole, octupole,
decapole, and duodecapole fields for correcting geometric and fringing field induced
aberrations at section H to 5th order. Multipole ML3 also contains up to duodecapole
fields in order to correct the aberrations of section L, as well as the aberrations imposed
by the immersion lenses. The remaining multipoles at section L (MLl, ML2, and ML4)
are imposed for correcting higher order chromatic aberrations up to 5th order. The effects
of the higher order multipoles are explained in the next section.

First order calculations of the map were used to obtain telescopic focusing at points
C, D, F, and (1. Thus, the terms (x,a), (a,x), (y,b), and (b,y) all vanish simultaneously for
the map at these points. Fringe field effects are always accounted for since they have an
efi'ect on the value of these terms.

The first order transfer maps for the first three sections can be evaluated and their

matrix product,

A = AL ' AI' Ari, (2'5)

gives the transfer map from point A to point F. The resulting energy and mass dispersion

terms are then

(x,5K)=(x,5K)L(5K,5K), —(x,x),(x,5x),, (2.6)

and

(L5...) = (105,), - (x,x),(x.5,, )” . (2.7)

respectively. In arriving at these results we have used (x,x)y =(x,x)L=-l for the
magnification of the mirror symmetric bend sections. The first-order result of the layout
gives the net energy dispersion (x, 50:0. After optimization the symmetry and telescopic

focus of both Section H and L causes (0.50 to vanish at each, and therefore, in the entire

53

system to obtain a firll achromat. The telescopic focus tends to make the envelope sizes
very insensitive to the detail of the initial beam ellipse.
The rest of the optics may be understood if Eq. (2.1) is applied to Eq. (2.6) to obtain

that

-K_L=1/(5K’5K)I: (“fig-)1.

KI] (x,a),.)u (x,X), I
Combining this with Eq. (2.7) we can obtain a value for the expected ratio between the

(2.8)

 

separation at point C to that of point F,

(x,a,),(x.x) =1__K_,__

. (2-9)
(x, a... )(xa X)” K}!

This factor implies that there is a loss in separation as KL becomes approaches K”. If
KL=KH then there is no net separation due to mass at the exit of the spectrometer. In
principal as long as KL is much smaller than K”, then only a small amount of mass
separation is actually lost.

As shown by Eq. (1.35), the emittance will grow as KL is lowered which sets limits on
the acceptance of section L. Also, there are limits on how high a potential can be
imposed on the ion source platform, thus limiting the kinetic energy, K”. The parameters
adopted for this system imply that we should expect about a 10% loss in mass separation.
This loss, however, is outweighed by the gain in resolution that results from getting rid of
the energy spread effect. Fig. 2.3 illustrates this point, where a series of x—a phase space
distribution plots is shown. Three sets of ensembles with masses m-Arn, m, and m+Am
enter the spectrometer with the same boundary-types phase space distributions. The
respective masses in terms of the relative particle coordinate, 5,, are -A, 0, and +A,

where A =1/20,700. We have assumed that the beam enters the spectrometer with an

54

 

 

aspect ratio of y.../x,,,=8 so that the beam has a 1 mm full width along the x—axis and
divergence lying between the limits of a,,.=:r:20 mr. In the y—phase space the initial beam
has :4 mm by :2.5 mr. Such beams are obtainable by the use of quadrupole multiplets

prior to the object slits as described in the literature [Wollnik91].

EndofsectionH
6M: —IA '0 TA 5K=ztA
fl—l (mm) 16511111)

 

 

 

 

0......-

r “ r
figs-:2
fl K.
<4..-____ .3--- *_
g? ~

-“~-

——.-”o'—- m. .—

 

 

 

 

 

 

 

 

 

 

 

.1510—
{r {A m
.\;\.:\
I .; =1 ‘-
. u 1: a
5 1: i E
g g g A=1I20,700
‘ 1’ i l
l r g |
. r; r: :
'1 t). 31 .1
1/1 1“ 1’
\ \/\/

=iA

R09

Figure 2.3 Mass spectra in x—a phase space for three masses of similar boundary-
type initial distributions. Going from the top-left to top-right plot shows the effect on
the mass spectrum at the end of section H when adding a random energy spread that
lies between (SF-A and +A. The dual-potential separation results in the spectrum
taken at the end of section L, which shows the effect of the achromatic correction.

55

The two adjacent plots at the top of Fig. 2.3 are snap shots of the phase space at the
exit of section H. In the top—left plot all particles have the same energy; i.e. 5(=0 for each
particle. In the top-right plot there has been a random distribution of energies imposed on
all particles, such that energies lie between &=-A and +A. The mass and energy
dispersion at the end of section H are (x,5()u=(x,5,,)u=23,000 mm, and it is obvious that
the energy distribution of the particles has a destructive effect on the resolution. The
particles of one mass will overlap the adjacent one by a maximum of 0.65 mm.

The plot at the bottom-left is a snap shot of the distribution at the exit of section L.
The particles still have the energy distribution imposed; however, the achromatic
condition that results by the dual-potential scheme has eliminated the energy dispersion.
Comparing this plot with the plot right above it, we see that the separation is smaller by
10%, such that the resulting mass resolution is m/Am=20,700.

To determine the size of the apertures for all the elements along the beam transport
line it is helpful to plot the beam envelope, or the maximum extent of the beam along the
transverse plane. A series of these plots are illustrated in Fig. 2.4 in which the s-axis is
projected as a straight line. Fig. 2.4(a) is a plot of the rays along the s—y plane. The rays
are defined such that their initial positions start from the limits of the y—b phase space.
This is also done for rays along the s—x plane in Fig. 2.4(b); however, for this plot it
seems that there are only rays of divergence from a point source. This is because the rays
that extend over position cannot be seen on the scale of the huge beam projections along
x resulting from the divergence of the beam. The elements are represented by the size of
the aperture of the corresponding elements. In the case of the sector magnets the ends

represent the gap between the pole tips.

56

 

(b)

X-VI-I

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) *
1%

Figure 2.4. Beam envelopes for the dual-potential spectrometer in the horizontal and
VCrtical planes. Plots are illustrated for the s—y (a) and s—x (b) planes for the effects

of? the transverse phase space on the rays. The last plot illustrates the effect of energy
dispersion along x.

There are some features with special significance that should be pointed out. The

57

integrated area under the x versus s curve within the dipole region is proportional to the
mass resolution [Wollnik71]; hence, the x envelope is maximized in the dipole regions.

The y envelope has been kept to a minimum throughout, which is the reason for
choosing an initial beam aspect ratio of y,Jx,,.=8. Part of the reason for having this is to
minimize the gap each dipoles to a rninirnunr. Also, the cross term aberrations between
the x—a and y—b planes at the multipole regions are reduced by a factor proportional to
the ratio between of x over y. These terms will cause a correction in one plane to cause a
distortion in the other [Yavor97a], thus it is critical that this ratio be sufficiently large.
The ratio has been kept at x,,/y,,.=86 between at multiple MH and is 9.2 at multipole
ML3. A combination of an increase in emittance in section L and the smaller x envelope
is what cause this ratio at ML3 to be ahnost an order of magnitude lower than at MH.

The plot in Fig. 2.4(c) represents the energy dispersion. Both divergence and
dispersion along the s—x plane are plotted for rays of varying divergence and energies
starting from a point source at A. Notice that the rays of different energies converge back
to the same position and direction as they exit the second dipole DL, showing that the
overall system is fully achromatic. From this plot we can estimate the maximum extent
of each ray of energy varying from that of the reference particle. The energy differences

are actually exaggerated by a factor of 10 for illustration purposes.

2.3 Correcting higher order aberrations

A considerable amount of effort has gone into determining higher order aberrations
and suppressing them with multipole fields. For determining the effects of higher order
aberrations the COSY INFINITY code system was employed to calculate maps of up to

5th order.

58

The fact that one can evaluate maps to arbitrary order is only one of the reasons for
choosing COSY. The code also allows the user to implement custom algorithms that
simulate other optical elements that are not in its own library. For this study it was
necessary to implement algorithms that calculate maps of some axially symmetric
structures that impose DC acceleration, such as those of immersion lenses. In the case of
an immersion lens, we simulate the potential for a simple gap lying between two tubes of
equal radii and of differing potentials [Geraci02]. Going to arbitrary order is particularly
important in the case of axially symmetric lenses, since their focusing strength comes
solely from fringing fields and, in the case of immersion lenses, the emittance can change
according to (1.35). The effects of higher order aberrations are potentially high and
having access to arbitrary order evaluations is useful in obtaining an accurate analysis of
a spectrometer that requires exceptional detail. Using map-based optics enables more
rapid system optimization
2. 3. I Aberrations at magnetic sector sections

Any optical system has geometric aberrations even when fringe field effects are not

considered [Wolhrik87c]. For a magnetic sector of large radius and large beam entrance

 

 

rrrultipoleaofl’ " hex-poison octupoleon decapoleon duodecapolaon
8
“a —_—_ __ - _ - ..
x 200mm
/ 1

 

 

Figure 2.5 Effect of applying each successive multipole field on the x—a phase space
plot calculated to 5th order at the end of section H. The effects from the y—phase space
are included.

59

angle of inclination aberrations can be especially large if not corrected. The plot of the
x—a phase space at the very left of Fig. 2.5, illustrates this. The snap shots for all the
plots in the figure are taken at the end of section H under successive application of
magnetic multipole fields at MI-I (see Fig. 2.2). These are all 5th order calculations.

The second order aberration clearly dominates before it is suppressed by a magnetic
hexapole field, as is done for the plot labeled as hexapole from above. For each
successive nth multipole that is turned on, the (n+1) order aberration appears to dominate
and requires that the next order multipole field be applied.

The field strength used for each multipole is approximately proportional to the
maximum extent of the distortion along the x—axis, starting with a 10 G/cm field
strength for the hexapole. We have assumed that the field comes from a circular
multipole having an aperture with 60 cm radius. In practice, a rectangular multipole that
more closely matches the beam envelope at position B will be used.

Besides the use of discrete multipoles, it is also possible to correct higher order
aberrations by shaping the edges of the dipole magnet. We have already imposed an edge
angle for first order focusing, but in addition the edges could be rounded relative to this
edge as is shown for the sector illustrated in Fig. 2.6. The first shape to notice is the
sector with no edge angles or curvature, labeled as simple sector. Then equivalent
positive edge angles are imposed at both the entrance and exit sides. Finally, a circular
curved shape is carved out from the exit side relative to the tilted edge. The rounded
edge should be at the side where the beam is largest in the horizontal plane. According to
Fig. 2.2 this has to be at the inside edges of the mirror symmetric dipole pairs. In order to

correct the second order geometric aberration of (x,aa)=-l34 m/rad2 at section H requires

60

’ Figure 2.6 Top view of a
magnetic sector in which
positive edge angles have
been imposed at both
entrance and exit
positions. In addition, a
round curvature shape

simple sector (small dasltrfisc) has been
- cut out at exrt to

)1 edge rormdrng correct the second order

+ aberration. The simple
7‘ sector (thin line) is that of
. the sector before edge

edge angle angles and curved edges

are imposed.

    

a round curvature with radius of 37 m. This curvature is barely visible from Fig. 2.2. On
the other hand, Section L requires one with a radius of 9.6 m to correct its second order
aberration.

The rounding of the edges tends to also correct other higher order terms. COSY
allows the user to specify any edge shape. In principle, this option can be used to search
for a shape that will suppress all higher order aberrations; however, the effects due to
fringing at the edges should probably be looked at in more detail before assuming that
this scheme will work in the real situation. This is because the effects on field uniformity
due to magnetic saturation may cause additional aberration effects for such a wide pole
tip. This topic is addressed in the next subsection.

Before going on, we should make some mention of the effect of the multipole fields
on other aberration terms. The on and off condition that the correcting hexapole field has
on the y—b phase space is shown by the two plots in Fig. 2.7. The smearing seen at the

boundaries is caused by the cross term (y,ab), which is suppressed by the hexapole field.

61

On the other hand, the cross term (x,bb) actually by about 0.3% aberration, which is not
very noticeable from the plots. Compared to the (x,aa), (x,aaa), and (x,aaaa) aberrations
most other aberrations are negligible in comparison and the corrective action on the
resolution of such terms evidently outweighs any growth of other terms. In the section
that follows, however, we will show that the combination of all aberrations set a limit on

how fine the resolution can actually become.

| b Figure 2.7 Effect of
hexapole turning on the hexapole

field correction on the y—b
‘ + t —— ——-~ + phase space at the end of
J y section H.

2.3.2 Obtaining homogeneous sector fields
Up until now we have assumed that the fields of any dipole are perfectly

homogeneous. This would seem to be somewhat of a reasonable assumption considering
that the required flux density at the pole tip at each sector magnet is only 3 kGauss;
however, perfectly homogeneous fields cannot be achieved in practice. Preliminary
magnet design calculations have been done to illustrate the sensitivity of the resolving
power to the field shapes. We have chosen to use the POISSON code system to
determine the field distribution along the x—y plane based on a multipole harmonic
analysis [Warren87]. An algorithm in COSY that allows the user to specify the field
distribution then calculated the map.

A preliminary cross section of the DH dipoles is assumed. Results of the 2D
calculations using POISSON are shown in Fig 2.8 in the form of field distributions. The

figure contains a diagram of the first quadrant of two H-type magnets along with the

62

fields predicted by the calculation. The permeability tables used are those that are listed
in the code for a type 1010 steel. A fit was used to determine the amount of current
necessary to obtain 3 kGauss at the center point. It is based on the assumption of a
uniform electrical current flowing through cross section of the coil region. The area
occupied by the coil is constant for all simulations as its horizontal width is decreased
and the height is increased. The pole half-gap stays constant at 5 em, but the groove in
which the coil is set into varies in depth from 0 (Fig. 2.8(a)) to 4 cm (Fig. 2.8(a)). The
thickness in steel between the top of the coil and the top of the magnet is kept constant to
keep flux in that region constant. Hence, any depth in the coil groove will require a
thickening of the top portion of the magnet, which increases the total weight of the sector.

For example, for zero depth the 60° sector will require 7900 kilograms of steel, while at a

(I) DH (all unit in cm)

\ 2
L' . ' A 7 ‘ ‘ 1n Willi) i

 

 

 

 

5

—.t

coils

(b)

 

 

 

 

 

 

61 — I
A 76 ‘1

 

Figure 2.8 Diagram showing the lines of magnetic flux for the design of the DH
sector of section H. The beam occupies a region that extends up to 43 cm in the
horizontal.

63

4 cm depth the amount of steel gets up to 10,100 kilograms at 4 cm.

The purpose of this exercise is to attempt to reduce the increase in field strength as we
reach the region of the coil. Fig. 2.9 illustrates this point where we have plotted the field
strength of the y—component of the B-field as a function of the distance fi'om the center of
the dipole. For the dipole with no groove we observe that the field rises continuously
until we reach the coil. As we make the groove deeper we see that the field is suppressed
sooner as we go out in x. This operation will also reduce the maximum field strength to a
value that is about 2% larger than at the center where x,,.=48 cm. According to Fig. 2.4
the beam is only expected to reach as far as x,,,--43 cm. At that position there is

comparatively little improvement by making any kind of adjustment at the coil.

The next step is to determine the effects on the aberrations resulting at the map if we

 

 

I l I l I
I T r r f fi fi fi
«1 -—’o.‘~
3000- """ 7:: -.
It
I
corlgrovedepth ‘1.
a —0— Gem 1‘
g --o—-O.5cm 1,
6 2900-- --l-- 2cm fl, “-
V "'O‘" 4cm :1"
an“ 2‘1.
1
'. l
2800-- l}-
27m 1 I . r ' I T I r I
0 10 20 30 40

 

 

x (cm)
Figure 2.9 Plots of the y—component of the B-field versus the x—position from
the center of the DH sector. The shape H-magnet is varied in depth in an attempt
to make the distribution more uniform.

apply the distribution for the dipole having the 4 cm groove. With all multipoles at MH
turned off, we calculate the map to 5th order at the exit of section H and evaluate the
phase space. The results may be compared against those of the homogeneous dipole by
observing the plot of the x—a phase space in Fig. 2.10. The inhomogeneous sector ends
up having all aberrations being almost 4 times stronger than those of the homogeneous
sector. This implies that even very small deviations from a homogeneous condition will
cause larger higher order effects. Henceforth, will be necessary to apply field corrections
by surface coils along the top and bottom poles. These types of corrections have been
used for various applications in the past with and show promising results [Wollnik72]
[Wollnik87e]. Since correction coils are more effective at imposing uniformity, there is
no need to go with the 4 cm groove H-magnet. This also serves to reduce the cost of the
magnets according to the scheme used above.

What is important at this point is that at least we have an idea of what type of field
distribution to expect and the effects on the mass resolution. The challenge will be in

providing a field that will have an integral uniformity with deviations in the field of

a
(60 mm)
non-homogeneous homogeneous

 

x
__ (70 mm)

Figure 2.10 Plot of the x—a for a 5th order calculation of a magnetic sector with
homogeneous field distribution and another with the field distribution predicted by
POISSON.

65

ABIBSIO'S. Also, it may be very useful to use types of steel that have higher
permeability in the range of field strengths needed for more uniformity. The quality of
the steel must be high to minimize fine structure in the field as well as the global effects
discussed above.

The design of the DL sectors has been considered as well. In Fig. 2.11 we show the
cross section of a dipole having a 6 cm gap and a horizontal region that extends between
+/-50 cm for the beam. The beam extends a maximum of about 16 cm in the horizontal
and less than 3 cm in vertically. There are non-uniformities visible by the lines of flux in
the region close to the coils. Imposing a greater degree of uniformity will require the
application of surface coils as in the case of the DH sectors.

Before going on, it should be mentioned that there are other methods used to obtain
wide homogeneous fields. For example, the so-called Purcell filter [Purce1155] requires a

separation between a pole tip and the magnet yoke obtain more uniformity. Another

lDL (cm)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
_L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

--r 36 H‘i
-‘

50 =—

 

 

 

 

Figure 2.11 Diagram showing the lines of magnetic flux for the design of the DL
sector of section L. The beam is expected to occupy a region that extends up to
about 16 cm in the horizontal.

method suggested by Halback is to introduce vertical slots in the steel to improve the
field uniformity [Nolen87]. Although these other methods may be used, surface coils
may still be implemented to permit fine tuning. Furthermore, surface coils can make up

for field strength dependent effects that other methods lack in.

2.3.3 The deceleration column
Some of the dynamics that are predicted by the calculations for the deceleration

column in Section I are explained here. For the map it is essential to determine terms
such as (do 5a and (x,x)l in Eqs. (2.6) and (2.7). The derivation of those equations relies
on the assumption that (x,a)l =0, and must be a condition that is imposed in the
optimization of the parameters. For good beam stability we also require telescopic
focusing of this section, such that (a,x)) vanishes simultaneously. Since potentially large
sources of higher order geometric aberrations may actually derive from the deceleration
column, then we examine the magnitude of the higher order terms.

It will be shown that it is possible to determine some aberration terms from the results
of ray tracing. This will be done from the initial and final positions given by the ray
trance result. We do this for the purpose of comparing the ray trance and map results
with some amount of detail, keeping in mind that it is more efficient to determining the
aberrations from the map-based approach. Modifications to the COSY code have been
implemented in order to calculate the map elements of single or multiple gap structures of
axial symmetry. COSY uses the potential, V(r,s), to determine the matrix elements via
integration of the equations of motions as described in Chapter 1.

Two methods exist for determining the potential. The first method relies on the fact

that a system with axial symmetry allows the potential distribution, V(r,s), to be

67

determined from the derivatives of the potential distribution along the s—axis (r=0)
[Reiser94b]. An analytical form of on-axis potential may be obtained from models based
on measurements as described in the literature [Hsi-men86]. One disadvantage is that the
analytical expression of V(0,s) is usually an approximation whose errors become
amplified as one evaluates higher and higher derivatives. Hence, one has to be extra
careful in calculating the higher order aberrations from approximate on-axis functions.
Another problem is that the parameters that can be specified with approximations are
limited in number and range of validity. This is true of gap models, such as the
IMMCAVl subroutine in Appendix B, where only the gap width and tube radius may be
specified as geometrical parameters.

The second method that is used to determine the potential off-axis is a more physical
model that approximates the system by representing the electrodes as a sum of discrete
charged rings. A Poisson solver determines the charge on each ring that is necessary to
satisfy the equivalent boundary conditions, usually the electric potential of the electrodes
[Geraci02]. Since the method is more accurate in determining the potential even close to
the walls of the structure, evaluating off-axis fields are more immune to errors at the
higher derivatives. Also, the method permits a realistic representation of the electrodes
by including effects such as the edge curvature of the tubes (see Appendix B). Once the
electric fields are expressed as the sum of the fields of charged rings, the field or potential
on—axis is represented as a sum of analytic functions. DA methods can then be used in
COSY to evaluate off axis fields to arbitrary order.

As an example, we start with the model for a single gap structure, or immersion lens,

as shown in Fig. 2.12. Ion trajectories come in parallel from left to right as they pass

68

 

 

 

 

 

 

 

 

 

Figure 2.12 Immersion lens structure with ions coming in with parallel trajectories

from left to right. The lines of potential along the center illustrate the fringing

obtained in the gap region.
through the fringing fields at the region of the gap shown by the lines of equipotential.
Notice that right as the beam passes the gap region, its envelope grows in size even as the
rays are getting diverted from the optic axis. This occurs because the first half of the gap
is diverging, while the second half is converging.

The results in Fig. 2.12 are obtained from a simulation using Simion 7. The values of
the initial and final radial positions, r; and r), respectively, are plotted in Fig. 2.13. From
them we can extract the value of the aberrations by applying a 5th order polynomial fit.
A comparison is made with the results obtained for the equivalent structure implemented
in COSY. We have used the immersion lens structure called ONEGAP in COSY, which
uses the method of charged rings to detemrine the potentials. The aberration coefficients
are listed in Table 2.1 in columns two and three, for COSY and Simion, respectively.
Notice that the amount of error in the coefficient gets larger for higher and higher order
terms. Notice, that if we apply r)=0.5 mm the accumulated error at rf between the two is
actually only 5.6%. The errors are mostly attributed to the fact that the curvature of the
tube at the gap regions is specified with an accuracy limited by the use of 5 mesh points.
The results, however, agree reasonably well in order to apply them to the map of the

spectrometer.

69

 

 

I

- - -Simion 7 4
-—COSY

 

 

-0.'02

0.60

r,- (M)

Figure 2.13 Plot of the
final versus initial radial
positions, rfand r),
respectively, of the
simulation in Figure 2.12.
A 5th order polynomial
fit is used to extract the
aberration coefficients
flom Sirrrion 7 results.
The curve obtained flom
applying the aberrations
according to COSY are
plotted for comparison.
The coefficients are listed
in Table 2.1.

Table 2.1 List of coefficients determined by numerical simulations.

 

 

 

 

order of coeflicient COSY SIMION 7 error

[units] coefficients fit coefficients [%]
1 (m) [m/m] 0.1663 0.1736 5.9
3 (x,xxx) [m/m3] -364.4 -281.9 23
5 (x,xxxxx) [tn/m5] -139900 -293100 110

 

 

 

 

 

 

Before going on, we should make note of the fact that these structures can be
arranged back-to-back to simulate multi—gap acceleration columns. There are some
restrictions, however, since the fields of separate structures are not allowed to overlap if
they are to be evaluated as individual maps. If two gaps of equivalent potential
difference are moved close to each other the superposition of the flinging fields will
effectively weaken the focusing power of the combined system. This effect will not be
reflected in the map if calculated under similar conditions. As a general rule of thumb,
the gaps should be separated by at least three tube radii in order to avoid an overlap
between the fields of each gap. Otherwise, the boundary conditions need to be redefined

to avoid errors.

70

The boundary conditions for multi-gap acceleration columns can actually be specified
with two of the subroutines added to COSY as part of this study. The subroutines
CUSCOLl and STDCOLI allow the user to specify the boundary conditions for multiple
column structures in conjunction with the subroutine ACCELCOLl, which calculates the
map. The use of these structures is described in more detail in Appendix B. These
accelerating columns are commonly used for boosting the energy of ions after extraction
from a region of ionization. They are also common to Tandem, Van DeGraff, and other
similar DC acceleration devices.

The accelerating column of the dual-potential spectrometer did not require
overlapping gap structures. Consequently, its map is calculated using two immersion
lenses of the type shown in Fig. 2.12. The simulation was also done with Sirnion 7, not
only for comparison, but also to obtain the beam envelope. Some development efforts
are still needed to make such graphical representations available in the COSY code
system.

The diagram of section I is shown in Fig. 2.14 as a cross section along the x—s plane.
The gaps are 1 cm in width and the radius along the column is kept constant at 5 cm. The
beam enters from the left at 100 keV and is decelerated to 29.9 keV in the first gap. The
map of the first gap is generated in COSY from the entrance of section I to the dotted
line, where it has parallel-to-point focus.

The map of the second gap goes from the dotted line on to the end of section I to
obtain point-to-parallel focus. The full map of this system gives telescopic focus with a

magnification of (x,x)1=-2.l35, and at (do &)=10 as the beam exits at 10 keV. A 1%

value of 5x before deceleration becomes 10% after. By symmetry the column also

71

imposes telescopic focusing in the y—s plane. The beam envelopes along both x and y
are plotted under the same scale in Fig. 2.15.

Finally, we compare the results of the phase space distributions obtained from Simion

 

 

 

 

x
(*5) z psocm4

Figure 2.14 Cross section of the deceleration column along the x—s plane of section I.
The beam comes in fl'om the left at 100 keV then gets decelerated to 29.9 keV in the first
gap and to 10 keV in the second.

 

 

  

 

 

 

 

16 I l : l l I r : l I
14: xm _
12- _
E 10~ _
v 8‘ -
5 . .
= 6- _
§ ‘ .
4.. s x . ‘
2" ; ‘\‘-” I 1
0 ‘ gap 1 (70.1 kV) gap 2 (19.9 kV)
0 500 1000 1500 2000
2 (mm)

Figure 2.15 Plot of beam envelopes along x and y. The dotted line represents the center
position of the gap for the labeled amount of deceleration.

72

 

. . D Sirnion7l
20. : + COSY ;

1 ‘33:- “'3:

 

a (nu)
. .9 .
b(mr)

 

 

 

 

v

38531'054'858'11'0'4 8
x(mm) HIM)

Figure 2.16 Phase space distributions in the x—a and y—b phase space given by
Simion and COSY.

7 and COSY in Fig. 2.16. At least in the x—a phase space plot, the efiects of the 3rd and
5th order aberrations are very clearly seen flom the boundary-type distributions. The
distorted ellipse given by COSY has a more pronounced "S-shape" due to the stronger
3rd order aberration flom that of Simion. It also may seem that there is an effect flom
extra drift in the COSY ellipse orientation; however, this is actually a result from the

difference between the calculated first order terms.

2.4 Purity according to the enhancement factor

In an earlier section we characterized the spectrometer based on its first order
spectrum and demonstrated its effectiveness at separating isobars at a mass resolution of
20,700, the results of which are presented in Fig. 2.3. Now we should like to characterize

the spectrometer in more detail by using a recommended standard.

73

We shall now introduce some concepts that are useful in characterizing the
performance of a spectrometer and apply it to the one calculated earlier. First the
following terms must be defined:

w - denotes wanted particles of mass m

u - denotes unwanted particles of mass m+Am

i - denotes beam before the separator

f - denotes beam after the separator

N - number of particles w or u particles accepted

C - concentration of accepted w or u particles

We assume that only unwanted species of mass m+Am make up the impurity. The initial

and final purity ratios are defined by,

r.- = N... IN...- (2.10a)
and

r, =NW/Nu, . (2.10b)

Usually one is interested in the concentration of wanted species after mass separation

defined by

C -—N-"-’——(1+1/r )‘l (211)
wt ‘ ‘ f ‘ '
NM+NIJ

But, rf is a function of r) and the mass resolution and may be determined flom,

EF=r,/r,.. (2.12)

where EF is the enhancement factor [Carnplan81], which is a function of the mass
resolution, Rm=m/Am. This enhancement factor varies with the emittance of the beam
and the shape of the distribution. If we know this enhancement factor then we can

always determine the concentration, C ”if, for some r,-. We will adopt the Gaussian

74

distribution as the one which best describes the phase space density and define the
emittance such that 95% of the particles lie within 4 times the rrns width (see Fig. 1.8)
under the phase space parameters defined at the end of Section 2.1.

The first thing that we shall like to determine is the maximum order that is necessary
in determining the effects on the performance. Again, one must be careful that errors are
not being amplified and making it seem as if we need more accuracy. In Fig. 2.17 we
plot the results of the transmission and EF as a function of the order of the calculation.
We have assumed that vertically aligned slits have been set at the exit of section L to
suppress unwanted beam, w. Notice that the transmission tends to level out after
applying higher than 3rd order aberrations. The effects caused by the aberrations that are
left after applying the multipole fields seem to have negligible effects after this point. It
seems that the best transmission expected for R...=20,000 will be about 93%.

The enhancement factor improves as higher order terms are corrected. The

enhancement tends to level out to a point where further improvements become minute

 

1.00 . . . . . 55
. -D—transmission
.50 m
0.98- -0— EF . 3-
~45 B
.5. 096. L a
. )40 §
0 94- +35 g1
.30 §
092- ~
‘ :25
0.90 . . . . 20

 

 

 

Md

1 2 3 4
Order of calculation

Figure 2.17 The transmission and enhancement factor as resolution of Rm=20,000
as a function of the order of the calculation.

75

after 3rd order. The effects of aberrations past 5th order seem to have little effect on the
spectrum and further corrections offer no improvement. The simulations are therefore
sufficient accurate at 5th order.

We also need to vary the separation between two masses in order to determine the
dependence of EF on R... Note that we take R... as being 1/5, here. Again, this result is
dependent on the initial density distribution, which we have assumed to be Gaussian.
The results are illustrated in the two plots in Fig. 2.18. Plot (a) is that of the final wanted
and unwanted concentrations as a function of R... assuming that the initial concentrations,
C...- and Cu), are equivalent. Plot (b) is the enhancement factor as a function of R,,,. Two
inset plots have been included to show the mass separations corresponding to both R,,,=
2,000 and 50,000.

By applying a phase space of zero width in the horizontal we are able to test the
contribution of aberrations to the final line width. Evidently, the resolution calculated
here seems to be very close to some fundamental limit. Right after turning on the 2nd
order aberrations the "line” spectrum looks no different that those with finite emittance of
the Gaussians distributions. The only conceivable way to go beyond this limit is to
further reduce the cross terms that arise from correcting the x—a phase space aberrations.

One dominant cross term that limits any further enhancement is the (x,atSr)
aberration. It grows with the 2nd order correction imposed by the hexapoles at MH and
ML3. Whenever the energy spread gets beyond :25 eV, the enhancement factor suffers
significantly due to this term. There were attempts to suppress this type of chromatic
aberration by applying multipole fields at MLl, ML2, and ML4 and fitting on the

reduction of the second moment along horizontal, <xz>. Although there was a slight

76

reduction to the second moment, a negligible improvement was seen on the part of the
enhancement factor. The only way to gain in the enhancement may be to reduce 5r flom

the source. This is one of the advantages of using the ion cooling system that follows the

 

'L0-
(L9:
(183
(L73
(16:
(15: * l
0.4-. ...—-—-
rial awe"
rial 1”“

(a)

 

fiactional concentration

0.1 :- ,4 If
(10 ‘H*“r. . , - , .
0 20000 40000 60000

Rm (“5...)

(b) .

d

i
1000;

 

 

 

80000 ' 100000

 

 

1

I

I v v v v ‘

r; : « . .
d 4

d I
.

 

 

 

 

 

101 1

 

 

 

. . . . . . . . . . . .
10000 1 00000
Rm (1/5m)
Figure 2.18 (a) Concentrations of wanted and unwanted species after separation as a
function of resolving power. (b) EF as a function of resolving power. Inset plot
shows the separation of two masses according to R...

77

gas catcher described in Chapter 1. We do not quote any numbers here for the
improvements that are expected, since gas catchers are a relatively new development and
are still under much research.

We summarize this section with a set of plots which show the expected mass lines of
the spectrometer system after all the corrections have been optimized for suppression of
hither order terms. The full emittance in the x— and y—planes are imposed to obtain the
mass spectra from lines of m/Am=20,000 as shown in Fig. 2.19. Plot (c) is that of a snap
shot at the end of Section L. We can compare this plot with two plots above this one
taken for Section H. In plot (a) we have removed the energy spread in the beam, while in
plot (b) we have imposed the energy spread to show the efiect of the energy dispersion
flom Section H. Without the energy spread imposed on the Gaussian distributions we get
very close to what is expected fl'om the 1st order calculation. The achromat condition
after Section L clearly improves the resolution of the system significantly.

Unfortunately, this may not be the end of the story when it comes to factors that affect
the mass resolution. We have not accounted for the possibility of formation of tails that
result from beam scattering with residual gases. This effect causes the formation of tails
that will also cause the enhancement factor to drop [Menat42]. The tails usually drop off
slowly as exponential as a function of momentum and can migrate far across the
spectrum. The dual-potential spectrometer, however, eliminates much of this effect
already. Since the effect grow in proportion to the vacuum of the system, it may also be
important to go with ultra-high vacuum systems. Also, it may be necessary to make some
studies on the effect of misalignments of any of the components. At this point we have

only seen the properties of an ideal spectrometer where everything is perfectly aligned.

78

 

 

 

 

 

 

 

o—u
N
b)

.3 -2'11'0

x (mm)
Figure 2.19 Mass spectra at two sections of the mass separator for m/Am=20,000. The
top most figure (a) illustrates the spectrum for a beam with no energy spread at the
focal plane of the first section of the separation. The next plot (b) shows what a
Gaussian distribution in energy with 95% of particles having energy between
AK/K=:l:5x10'5 (:15 eV at 100 keV). The last plot (c) demonstrates how the achromatic
character added by the section after deceleration can improve the resolution.

Furthermore, one also has to keep in mind that ripples and other power supply
instabilities will be magnified by a factor proportional to Rm and providing feedbacks

systems for stability will be an important part of the overall system design.

2.5 Issues related to beam matching and the pie-separator

This chapter shall end with a discussion related to sections of the rare isotope
accelerator that lie before and after the isobar separator. This is relevant since those
sections affect some aspects in the design of the separator. We shall also point out how

possible alternatives to the pre- and post-separator sections will affect the design.

79

2.5.1 Obtaining the required aspect ratio

Mass separator performance will always trace back to the beam characteristics of the
emitting ion source. We have assumed that the extraction has axial symmetry and that
the x—a and y—b emittance are equivalent. A beam of 101: mm-mr emittance area for a
100 keV/250 amu implies that there is a 0.0093 mm-mr normalized emittance. Although
this is a reasonable assumption for most common ISOL ion sources, beams extracted
from electron cyclotron resonance (ECR) type sources must be considered as a possible
option. Such sources are good for providing higher charge state; however, they tend to
have emittances that are higher by at least a factor of 3. Since the separator is designed to
accept a maximum divergence of am=.1:20 mr, the beam aspect ratio would have to shifted
to y,/x,,,=8/3 for a beam width of 3 mm in x. Consequently, the mass resolution would
sufier by at least factor of 3 if the transmission were to be conserved. In order to obtain
different aspect ratios, however, there will be the need to design a beam optical system
that can vary the output y..,/x,,,. Beam matching with the separator will require that it also
yields upright ellipses (a,= a,=0), fl=1.6, and a value of ,6, that varies flom 0.025 to
0.075.

One possible design solution is shown in Fig. 2.20. This sextet system is reminiscent
of the one in Fig. 1.7. It has the same overall length, but the triplets have been shifted

away from the center by 11.5 cm. This sextet is different in that it has magnifications of

Mx=-1/J8 and My=-l. The plots of the initial and final ellipse show the effect on the
x— and y—phase space. We have assumed that the beam is circular at the entrance with
flx=A=L6 and ax: ay=0, and its telescopic properties insure that the ellipses will be

upright at the exit. This system has had to lose the symmetry in the excitation of the

80

quads compared to those of the sextet in Fig. 1.7; however, the values of the excitation
difl'er by less than 20% from the symmetric case. We have listed the excitation values in
Table 2.2 for direct comparison. We should note that it is possible to shorten the outside

quads and lengthen the middle quad a bit for each triplet value is similar in magnitude.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CD :\— F) T Q
._-a/ / I N_,//?
l <21 E a : 92 a6
S
l
|

 

Figure 2.20 Sextet with magnifications ofM =-1/J§ and M,=-1. Dimensions are

similar to those of the sextet in Fig. 1.7, except that the triplets have been shifted
outwards from center by 11.5 cm

Table 2.2 List of electric quad excitation voltage values in kilovolts.

 

 

 

 

 

 

 

 

 

 

quad Fig. 1.7 Fig. 2.20
Q1 -0.7415 -0.7576
Q2 1.309 1.152
Q3 -0.7415 -0.6578
Q4 -0.7415 -0.8918
Q5 1.309 1.502
Q6 -0.7415 -0.6484

 

 

81

In order to obtain an aspect ratio of y,,./x,,.=8 it necessary to use two similar optical
systems in tandem to obtain M,=(-1/J§ )(-1/J§ )=1/8 and A4,: (-1) (-1)=1. Finding the
solution of a single sextet that obtains Mx=-1/8 has not been possible using the
dimensions of the quads shown; however, further studies might yiekl such a solution.
The advantage of using a tandem system is that it allows some manipulation of the aspect
ratio while still retaining telescopic focusing. This ultimately gives the system more
options in tuning the system in order to match the beam to the separator.

One other important thing to keep in mind is that after the mass separation, it will be
necessary to bring the beam back to an aspect ratio of y.../x,,,=l under upright ellipse
focus. Otherwise, the rest of the rare isotope accelerator must be designed to accept
aspect ratios different fl'om unity, which may not be very practical The ideal thing to do
is to place a similar matching system but in reverse right after re-acceleration to 100 keV.

2. 5.2 Choosing the scheme of separation
The solution for the dual—potential spectrometer as introduced above depends on a

deceleration from 100 keV to 10 keV between the two magnetic separations. The system,
however, can just as well be operated in reverse, in which case the beam must come in at

10 keV fl'om the side of the source, be separated in momentum by section L, and then be
accelerated to 100 keV for the energy correction flom section H (see Fig. 2.2). The
diagram in Fig. 2.21(b) depicts this situation. The diagram in Fig. 2.21(a) depicts the
current design scheme. A third situation is depicted in Fig. 2.21(c) in which the beam
comes in at 10 keV, is accelerated to 100 keV to go through the same system in (a)
followed by a deceleration back to 10 keV. The differences lie in how other essential

sections are floated at high voltage (HV) on top of an isolated platform, or HV Deck.

82

 

HV Deck (+90 kV)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) ' '
decelerate accelerate
Source H Sect H l_. 100 keV to 10 keV to _,
100 kV 10 keV 100 keV
accelerate i

 

RFQ
AK=-90 k v
—v to 380 keV m 000‘“ decelerate l

 

 

 

 

 

 

 

 

 

 

 

 

 

. ........ EYPSCJLQPQ 1912---.
(b) 1 I
' accelerate ~ decelerate
30““ '4 5°“ L I.” 10 keV to m 100 keV to _.
10 "V , 100 keV 10 keV

 

 

 

 

 

 

accelerate RF Q g
"F “(=0 keV to T'"| bunch decelerate l"""
470 k eV ‘ 0.0015c 7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-----_-.Hx_9991c.i-.9.9.Isy.> ........
(e) '
accelerate decelerate
30““ 10 keV to m 100 keV to ..l 5°“ L I...
10 RV 100 keV 1 10 keV
accelerate l
"'i AK=0 keV to bunch 10108 15 A decelerate ]'—>“'
470 keV ' ° , ,

 

 

 

 

 

HV Deck (0 to -470 kV)

 

Figure 2.21 Various schemes of source extraction, separation, and post-
acceleration. Preseparation and beam matching have been left out for simplicity.

83

 

Every section that requires isolation has been labeled and encompassed by a dashed line.
It is assumed that there exists a DC acceleration column at each boundary where the
beam crosses the dashed line. For simplicity, we do not show the beam matching
systems discussed in the previous subsection.

It is also necessary to consider the first stage of RF acceleration that follows right
after the mass separator. That section will consist of a buncher and a radio frequency
quadrupole (RFQ) accelerator that accepts beams of velocity 0.0015c. Since ions of
different mass will have different velocities coming off of the separator, it will be
necessary to apply DC acceleration before the ions enter the RF accelerator. This
requires that the RF accelerator system be isolated at high voltage. Below the boundary
that depicts the HV deck of the RF accelerator we show the necessary range for the
potential on the platform considering a mass range between 6 to 240 amu.

A preliminary evaluation has been made to assess which scheme is the most practical.
For each scheme we factor in some of the technical difficulties and cost. The following
factors favor the use of scheme (a) in Fig. 2.21:

1. Sect L is physically smaller, (see Fig. 2.2). More floor space is available if

section L is isolated instead of Sect H.

2. An additional 90 kV would be necessary if using scheme (b) or (c) to match the
accepted RFQ velocity. Considering that the RFQ system may require power
levels on the order of 50 kW favors using lower isolated potentials.

3. The distance between source and separator is expected to be at least 20 meters

long, which makes it necessary to consider effects during transport. The cross

84

section for scattering with residual gases is proportional to the inverse of the
velocity, making such events at least 3 times less probable with 100 keV ions.
Also, the higher emittance at 10 keV would make the beam more susceptible to
aberrations fl'om focusing elements.

4. Fluctuations of the voltage at the ion source on top of 100 kV are less noticeable
than at 10 kV. This can be important if there is any energy resolving devices. For
example, to resolve a :5 eV energy spread at 10 keV requires 10 times less
resolving power that for a 100 keV beam.

Whatever scheme is ultimately adopted will affect the isolation of both the source and RF
acceleration. There are still some development efforts that may go into the final

determination.

2.5.3 Considering some aspects of the pre-separator

Before concluding this chapter we should consider some minor aspects of the pre-
separator. The current layout of the RIA facility has the production targets several meters
below ground level [Savard01]. The isobar separator will be at ground level along with
the rest of the post accelerator system. This requires that the beam be bent upward and
then horizontally again as shown by the scheme in Fig. 2.22. This section makes up what
is called the preseparator system.

The scheme shown in the figure assumes that there will be no horizontal bends
imposing momentum dispersion. This entails using electrostatic deflection plates as the
one shown for the bend that sends the beam up towards ground level. This bend may be
used to limit the maximum energy spread of the beam. This will be important in

eliminating energy tails that form at the extraction region of the source where vacuum

85

tends to be poor [Camplan81]. An energy resolution of K/AK>1000 would suffice the
best here (i.e. accepting a :50 eV energy window).

The first dipole magnet should bring the beam back to the horizontal plane. It will
allow the option of sending of beams containing other isotopes of interest to alternate
channels where they may be directly studied or colbcted for medical and other
applications. A resolution of about R,..=1000 may be sufficient for this purpose. The
bend should avoid adding any dispersion that will interfere with that of the isobar

separator. This separator should only impose (y, 5,.) and (y,&) dispersions to the beam

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 
    

 

 

7

/. . m
76 /‘ , *
// 2
Figure 2.22 Conceptual design of preseparator for the RIA facility. The beam must be

transported from below ground level to the isobar separator and rare isotope accelerator
above ground.

 

 

 

86

and avoid any dispersion in the x—plane. This will avoid any chromatic interference
with the isobar separator and allow the beam to keep a more definite structure in the

horizontal plane. The rest of the beam transport in the horizontal plane should either be

electrostatic or achromatic.

87

Chapter 3

APPLICATIONS WITH ACCELERATING RF DEVICES OF AXIAL SYMMETRY

In this chapter we shall focus on aspects related to the RIA driver accelerator,
although some are applicable to the rare isotope post-accelerator as well. The first
section of this chapter will discuss the implementation of axially symmetric electrostatic
devices with time-varying fields into the COSY code system. Such devices are necessary
not only for acceleration, but also for preserving longitudinal properties of beams with
bunch structure. Some numerical examples are provided that may serve as a reference
for calculating maps of such structures. The next section will discuss the application that
requires the optimization tools and higher order capability of COSY for the design and
simulation of multiple charge state beam transport systems in the RIA driver. It will also
cover some background about the design of the driver system and reasoning behind the
use of multiple charge state beams. Finally, we shall give an overview of the charge

stripping calculations, since they provide the basis for the design.

3.1 Axially symmetric devices with time-varying fields

Electrostatic devices having axial symmetry play a vital role in acceleration systems.
In fact, axial symmetry is used almost exclusively in all accelerating structures. There
are some exceptions, such as in the case of RFQ accelerators, which have midplane
symmetry. This type of accelerator, however, is mainly applicable at low ,6
(approximately fl=0.001 to 0.05) where the electrostatic quadrupole strength is most

effective for very heavy ions of large mass to charge ratio, m/q. On the other hand,

88

structures with axial symmetry having static or time-varying fields are used throughout
the spectrum of velocities.

The previous chapter covered axially symmetric structures with DC acceleration and
demonstrated that maps of such structures could be calculated to arbitrary order with
COSY. This chapter will discuss how the same can be done with structures having time-
varying fields. The power of using map optics for optimizing the design of a system will
be demonstrated with a particular application at the driver linac system. We shall use the
term RF structure or device at times since the velocity of heavy ions and the practical
dimensions of accelerating structures demand that the fields vary at RF flequencies

(approximately between 1 MHz and 1 GHz).

3.1.1 Time-varying fields in COS Y
The algorithms in COSY are very well suited for applications in beam dynamics in

which energy conservation applies. This is especially useful in repetitive systems, such
as in storage rings. As pointed out through (1.221), however, much of this capability
stems fl'om the assumption that the variable, 5:, is a constant of the motion, which tends
to limit the way maps for RF structures are evaluated. The only known subroutine that
calculates maps of structures with time-varying fields in COSY is one called 'RF', for RF
cavity. It allows the approximation of a cavity by applying a kick in the electromagnetic
potential over some infinitesimal section of s. It also allows the kick to be variable along
the x—y plane. Unfortunately, it is not straightforward with such a model to make a
correlation between the simulation and the geometry of a realistic RF device. To go
beyond this limitation the program needs to be able to evaluate systems that take up a

finite amount of space along s and account for the geometrical dimensions of the

89

structure in some detail. In particular, one should have the option to specify the time-
dependent field distribution of the system. This requires that the equations of motion
have an s dependent form of 5" as was demonstrated by Geraci and others by the

derivation of the equation [GeraciO2],

6,} = 2K n, [aa'+bb'+££fll—] (3.1a)
leopo
where the quantity,
N, - p‘" l (3.1b)

 

”‘0 $12 +122 +(p../po)2 +(mC/po)2

is the so-called relativistic factor. Here, p, is the longitudinal component of the
momentum vector, p=( p,, p,, p,), for any arbitrary particles in phase space. [(0 and po are
the reference particle momentary kinetic energy and magnitude of the momentum,
respectively. The rest of the constants and variables are the same as those defined in
section 1.5. Eq. (3.1) is incorporated into a modified version of COSY such that it
replaces Eq. (1.221). It is important to keep in mind that in deriving Eq. (3.1) only the
time dependent interaction of the electric field is accounted for and time-varying
magnetic fields are neglected. This is generally a reasonable assumption, considering
that axially symmetric induced magnetic fields are very weak in comparison to those of
the electric. In realistic situations there are some stray electric and magnetic fields that
are induced. Some of them stem flom the electrical flow through the supports inside the
structure [OstroumovOl]. Current RIA based studies have demonstrated that some of
these fields carry a dipole component that steer the beam, especially for quarter-wave
resonators at frequencies as low as 115 MHz. Such effects are, however, mainly

perturbative in comparison to the effect of the axially symmetric fields. They are beyond

the scope of these studies, and shall be neglected here so that we may concentrate more
on the details that are of primary importance in the dynamics of the beam.

The added capabilities in COSY allow the user to simulate effects flom axially
symmetric one-, two-, three-, and four-gap structures, as described in Appendix B. A
sinusoidal time dependence on the field has been assumed for any structure, although the
user can program other functions if necessary. The one-gap model is conceptually just an
immersion lens with the potential drop between the tubes varying sinusoidally. The field
distribution for calculating the map of this structure may be evaluated through either the
method of evaluating derivatives of the approximate on-axis potentials [Reiser94b] or by
using the more accurate method of charged rings [Gerac102]. This also applies for the
two-gap structure. For all other structures only the method of charged rings is applicable.
Aside flom being able to specify the maximum potential and phase of the accelerating
tubes, it is also possible to specify other parameters. Some of these parameters include
the gap widths, spacing between gaps, radii of apertures, and dimensions of outer cavity
walls. The parameters may be varied to optimize beam properties along the x—a, y—b,

and l—& subspaces as will be discussed next.

3.1.2 Properties of the longitudinal phase space
A brief overview of is given here of the expected values for the longitudinal subspace

map. Also included is a review of some of the relevant quantities that are useful and any
units of conversion that apply to the longitudinal phase space. Since the effects of the
longitudinal phase space on its respective ellipse are slightly different than those of the

transverse phase space, it is helpful to start with some discussion of the Twiss parameters.

91

It is usually more common to find longitudinal phase space expressed in terms of time
and energy difference, relative to the reference particle. In COSY the choice of units are
those of l and (55 which are proportional to time and energy spread as pointed out in
(1.6). With the use of these coordinates one still needs to define some ellipse shaped
boundary that is based on the beam distribution in longitudinal phase space. The Twiss
parameters of this ellipse are related by the relation,

13171-012 :1, (3-2)

and the maximum extent of the boundary along the 1—5( plane (see Fig. 1.5) are given
by.
1,, 4% and a,” = y,e, . (3.3)
Here, 8; is the longitudinal enrittance in the l—& plane and has units of meters. In these
units one often refers to the total bunch width as the quantity 21,, and the total energy
spread as 251g... A quick review of some useful conversions to obtain these results in
different units is now in order.
All quantities are specified relative to those of the reference particle, such that At is
the difference in time-of-flight, As is the difference in length along the optic axis, A]? is
the difference in velocity over c, Ap is the difference in momentum, and AK is the

difference in energy between some particle and the reference particle. According to the

definition of the canonical variable, 1, in Eqs. (1.6) the relation,

70
A1=—At-p =As—-. (3.4)
° (1+yo)

may be used to convert between units of the first coordinate in the plane. For the other
coordinate we can use the relativistic forms of momentum and kinetic energy to arrive at

the relation,

92

Afl 1 AK 1 Ap
= = —. (3.5)
flo 70(70 +1) K0 7oz Po

Thus for example, to convert a into units of time-of-flight and energy one needs to
multiply by the factor KoI/ro. A typical longitudinal emittance is expressed in ns—keV/u
which would further require that this quantity be divided by the unit mass, A.

Instead of using units of time, one can also use units that correspond to the RF phase
shift, Aqr, for some frequency, v. This phase shift can be calculated by Apzw-At,
where ar=21tv is the angular frequency in units of radians. The bunch width may then be
specified as 21..., 2At.,., 2As,,., or 215(1),... while the relation in (3.5) can be used to obtain
velocity, energy, or momentum spread in the other variable. With this bit of useful
knowledge we return the discussion to the longitudinal subspace.

One distinct feature that is different from the transverse plane derives from the
transformation of the sigma matrix. Since the longitudinal phase space is not dependent
on the energy of the reference particle, the inverse transformation no longer requires the
application of some scaling factor, such as pop/poi in the transverse subspace. The final
Twiss parameters are evaluated from the initial ones by the following equations resulting

from the transformation of the sigma matrix of the type shown in Section 1.6:

r... = now. .6. )2 - 2a... (6. .005. .6. ) + ,6,,.(6,.1)2 (3.6a)
82,, = -7,.0 (1,5,. )(6,. ,6, ) + at,o ((1,1)(5, . 6, ) + (1,6,. )(6,,1))- 0,, (1,1)(6, ,1) (3.6b)
16)., = 71.009611 )2 - 201.0 (1.5.. X“) + fire (1.1)2 (3-69)

We shall now like to consider the first order matrix elements under the action of a
drift of length, Lo. To begin with, a particle will not experience any change in energy so
that (5},5r)=1, (l,l)=l and (550:0. The relative shift depends solely on the velocity of

the particle relative to that of the reference particle by,

93

 

Al-—(t-to),&‘ =_Lo(_l_.__l_ ...LzLOAEL, (3.7a)

0+7.» 19 130 (1+ro) 19. (H70)

and with the use of (3.5) yields the form,
= L0—— +251) =,(l 5 K )5. (3.7b)
Hence, the longitudinal submatrix for the drift simplifies to,

 

(1.0 (1.6.)]_1 L0

A = — 2 . 3.8
KL") [(5,4) (6,,5,) o (“I”) ( )

The values of the matrix can then be substituted into (3.6c) and we obtain from the first
equation in (3.3) that the half-bunch width varies with the initial twiss parameters

according to the relation,

 

710 2 2am
I _ O 3.9
..=(Lo> 1/£[(ro+1>‘ (L.)- ———(Lo)+(70+1)+fi..o] ( )

This function defines the envelope within which the bunch resides in throughout
propagation along the drift. The equation inside the square root is that of a parabola
which opens upward. The minimum is always greater than zero and occurs at
LG =03... (ya2 +1)/ no to give I", =W . The minimum may occur at a positive or
negative value of Lo depending on the sign of am.

It should be noted that for the subspace in the longitudinal plane a drift does not
imply that there is a field free region. It only means that there is no time dependent fields
and that the Eqs. derived in (1.15) under the arguments of symplecticity still apply. As
far as the topic of map elements under time-varying fields is concerned, we shall only
study those having axially symmetric properties. Monomials containing 1 or 5‘ will not

necessarily vanish; therefore, the transformation of the type described in Appendix A

94

would no longer give the same symplectic relations. Although we do not go into the
details of the canonical transformation, it is important to mention that such relations can
be useful in checking the validity of some map elements. Although one can use the
symplectic approach to check accuray, there is little hope that it will lead to better

performance in evaluating all the coefficients of the map.

3.1.3 Map calculations with RF devices

This section will illustrate the numerical results from a simple model with time-
dependent fields. The model consists of only one two-gap structure that is set at the
center of two equivalent drifts at either side. Between the starting point and the midpoint
of the first encountered gap is a distance, L. As shown in Fig. 3.1, the system is
symmetric about the center point of the tube that has a potential varying as,

V = V0 cos(at + qr) (3.10)

where Vo is the maximum potential. The tube is centered inside a grounded cavity
structure that also has axial symmetry. The voltage characteristics and dimensions of the
system are specified through the input parameters of a subroutine, called TWOGAP
described in Appendix B. The parameters used in this example are listed in Table 3.1,
where the parameters Vo, qr, and L are vary.

The gap-to-gap distance has been evaluated based on the velocity of the reference

 

 

 

Figure 3.1 Two-gap structure system symmetric about the center.

95

Table 3.1 Parameters used for TWOGAP cavity in Fig. 3.1.

 

Frequency (v) 350 MHz
Vo (varies)
‘1’ (varies)
Gap-to-gap distance 17.1 cm
Width of both gaps 2.1 cm
Inner radius of tube (RI) 2.5 cm
Outer tube radius (RC) 5.0 cm
Aperture radii (RI) 2.5 cm
Radii of curvature (RE=RD) 0.5 cm
Cavity radius (RRES) 7.1 cm

 

particle and the flequency of the cavity. For this application we need the particle to
experience the same polarity and magnitude of the field at both gaps. This implies that
the cavity must undergo a (n+l/2)7t shift in p as the particle travels flom one gap to the
other, where n is an integer. If we assume that the velocity does not vary appreciably
after the first gap then the gap-to-gap distance is simply given approximately by,

MGM/2) (3.11)
where hv=c. The gap width has been taken to be one eighth of this distance.

The parameters assumed for the incoming beam are listed in Table 3.2. These are
required of the application to be discussed in a later section. Notice that we have
assumed that the longitudinal ellipse is initially upright; thus, according to (3.9) the
bunch is expanding with s as it approaches the cavity. Since the transverse phase space

effects will be neglected for now, we assume that ex=£y~0 for these calculations.

96

Table 3.2 Properties of the incoming beam for the model described by Fig. 3.1.

 

 

 

 

 

 

reference particle enggy/A 85.3 MeV/u
unit mass (A) 238
charge state (qo) +90
14.7 um
9’ (20 keV-ns/u)
at 0
21... 4 mm

or 2A¢,,. 8°

 

 

 

 

The effects that the oscillating field will have on the bunch are heavily dependent on
the phase of the cavity. To show this dependence we take the case of L=13.2 m and
Vo=2.2 MeV being constant as the phase varies from —l80° to 180°. The plots in Fig. 3.2
show four different quantities as the phase is varied relative to the phase or: are. The
phase a is defined as the point where a minimum in (5d) occurs, which is at -53.4°
shown by the upper plot. Notice in the same plot that (1,152) also goes through a minimum
at this phase. This is characteristic of system with symmetry along s.

The bottom plot shows the flactional energy gain of the reference particle. The
maximum energy gain occurs at A¢=90°, and signifies that at this phase the tube takes on
the maximum negative potential as the particle is in the region of the first gap.
Conversely, maximum energy loss occurs at A¢=-90°. Although not obvious by the
plots, what is happening between Aer—90° and 90° is that the particles at the front of the
bunch are losing energy relative to the ones at back. This is referred to as bunching in the
sense that particles in the front had a higher energy than the reference particle, and
therefore, arrived earlier. The rate of expansion of the bunch is either decreased, or if V0
is high enough, the bunch will actually begin to contract. Going from expansion to

contraction is equivalent to stating that to changes sign from negative to positive.

97

Outside of this phase region the opposite effect occurs and we call it debunching. Note
that (5d) remains negative at the bunching region of the phase. Since this is a general
property, it is useful to impose the conditions (5ol)<0 and (Koy-Ko,o)/ Ko,o=0 when
searching for g)... It should be also be pointed out that the "bunching strength" of the
system is actually proportional to this quantity, and we shall refer to (551) as the
"bunching strength" of the cavity.

One interesting aspect of axially symmetric devices in DC mode is that they always
yield a positive refractive power, llf, regardless of the polarity of the field [Reise194c].
This property, however, does not translate over for the case of oscillating fields. As seen
by the second plot in Fig. 3.2, the reflactive power actually varies with the phase of the
cavity. What we actually get is a reflactive power that is proportional to (561), and
consequently, there is maximum defocus at maximum debunching. This is an
unfortunate quality, since a transport system requires bunching throughout in order to
preserve beam stability in longitudinal phase space. The ideal situation is to get both
focusing and bunching, simultaneously, as in the case of a well tuned RFQ. From the
plots we can conclude that for a linac with axial structures it is more ideal to approach
A¢=90° flom A¢=0 without losing too much bunching strength. Unfortunately, there is a
trade off and one must settle for something in between that preserves the beam stability.
Furthermore, there will be a need for refocusing structures to offset the defocusing effect.
Applications related to beam stability of more complex systems, such as extended
sections of linear accelerators, would be ideal for this type of implementation in COSY.
Instead, we shall redirect the discussion to aspects that are related to higher order effects

with cavities.

98

(1.6.)

 

 

 

 

 

 

 

 

 

 

 

3 —1:1-—' ' ' ' . ' - 010
0 002 - (K0! ' K0.0)/K0,0 + l/f: -(a,x) .
bunching] 0.010
1.
defocus
3’ 0.001 - ’0 005
M 0
>2. . :
¥ ‘5
I 0.000 0.000 n
S A
35° r 1;:
-0.001 - l--0,005 V
l
'0'002 ' --0.010
-180 -9'0 ' 0 f 9'0 180
A¢= ¢'¢,,

Figure 3.2 Relative phase dependence of the quantities (1,50, (551), l/f, and (Kay-

Ko,o)/ K00.

The most straightforward method of showing higher order effects in longitudinal

phase space is to extend the bunch throughout the RF cycle. We shall define the length,

Al(2rt), as the distance that the full bunch occupies throughout one period of the RF

cycle, which is given approximately by

99

M 271’ 6160616.. 70(1'1'70)
z———— l — . 3.13
( ) V(1+70)2[ + an» J ( )

This assumes that we initially have an upright ellipse, and that after a sufficient interval
of time 1.31.. Here,

I, =—a,JeT/7, (3.14)
in analogy with the subscript convention used for the x—a phase space in Fig. 1.5. This
approximation applies well within a few percent for the ellipse defined in Table 3.2, and
we adopt those parameters for our initial conditions in this example.

It is also helpful to have an idea of how far the bunch can propagate before the full

bunch width becomes the size ofAl(211:). Applying Eq. (3.7) this distance is found to be

(1+ ro)’ Al(21t).

L =
(2”) 5,,”I 2

(3.15)

 

Furthermore, we define the ratio,

r, = 21,, max). (3.12)
which gives the fl'action of the RF cycle occupied by the bunch width. For the ellipse

here we find that L(2n) is 53 meters for $40.5.

The symmetry of the system in Fig. 3.1 makes it possible to reorient the phase space
ellipse with the cavity so that it makes a complete flip by the time the bunch reaches the
exit of the system. It requires that the relative phase be set at AM in order to bunch
without any gain in energy by the reference particle, and that the value of V0 be set so that
the energy and time variable magnifications, (555V) and (1,1), respectively, yield a value
of -1. This rebunching action is synonymous with telescopic focusing in transverse
phase space as it has that (5(,l)=(l,5()=0. We solve for the value of Vo for a series of L

values and plot the results in Fig. 3.3. Notice that as the size of the incident bunch at the

100

cavity grows, while the amount of cavity potential necessary for rebunching goes down
exponentially. Clearly there is much to gain in terms of reducing the required RF power
for rebunching. On the down side, we see from Fig. 3.4 that the there is a price to pay for
allowing the bunch to increase. The phase space plots in this figure have been evaluated
to fifth order flom the values in Fig. 3.3. Notice that the aberration effects grow with
bunch width. The aberrations effectively increase the longitudinal phase space and may
ultimately result in halo formation as the filamented phase space propagates.

Clearly, there are a number of trade-off factors to be considered in rebunching and
acceleration. The distortions in the phase space ellipse, whether transverse or
longitudinal, translate into emittance growth. At this point we have not even considered
the effects of changing geometrical factors of the cavity or the linearity of the time-
dependence of the field. Such factors are restricted by certain technical aspects of

constructing RF devices and are beyond the scope of this study. Instead, the discussion

 

 

 

 

 

m ' f T ' t ' r ' f
1- 0.5
L
)- 0.4
- 0.3
p ..‘t
+0.2
1
~ 0.1
I ' I ' I ' r - 1 0.0
0.1 0.2 0.3 0.4 0.5

L / [(211)

Figure 3.3 Cavity potential Vo necessary to obtain upright ellipse at exit of system
given length L before the first gap.

101

 

l
wI—i
l
l
.354.)
l
I

_ 3...)...

i

1

 

r302 ! r;=0.35 j“ r;=0.5
A1(2n) ——.J

Figure 3.4 Resulting phase space plots for each corresponding r) as evaluated to
5th order by COSY.

 

 

 

 

 

below concentrates on a particular application in which cavities are used for bunching a

beam of multiple charge states.

3.2 Applications with multi-q beans

The RIA driver linac is to be designed with the capability of delivering high power
beams of ions fl'om the complete spectrum of masses flom the periodic table. Except for
a few room temperature RFQ accelerators at the low energy section, the entire linac is
based on current state-of-the-art superconducting (SC) elements. With over 400
independently phased SC cavities ranging in flequency flom 58 MHz to 800 MHz, the
driver is expected to deliver up to 1.3 GV of RF potential [Shepard99]. This translates
into ~900 MeV protons and ~400 MeV/u uranium ions, when considering independent
phasing for variation of the velocity profile at each mass. The maximum energy per
nucleon becomes limited at lower values for elements of higher Z due to the increasing
difficulty in stripping electrons away. A diagram illustrating a simplified diagram of the
driver is shown in Fig. 3.5 for the acceleration scheme for uranium.

A number of factors affect the decision to use SC technology, most of which stem
from the high power requirement of the linac. The ultimate goal is to deliver up to ~400

kW in CW to multiple targets. SC cavities are ideal for CW operation, and their

102

relatively short, high- gradient designs provide very strong longitudinal focus and high
transverse acceptance. This will ultimately lead to reduction in halo formation from
higher order aberration effects. Furthermore, CW beam minimizes transient effects in the
target that would otherwise occur when using pulsed beams, such as from synchrotron
based schemes or room temperature linacs in pulse mode.

Probably the most unique feature of the linac is that it will accelerate multiple charge
states (multi-q) beams. This feature allows charge states that would otherwise be
diverted to a beam dump, to be accepted by the linac for acceleration. It also reduces
unwanted radiation at parts of the facility that would otherwise require a considerable
amount of shielding. It is estimated that the losses fl'om not accepting multi-q beams
would have to be made up by requiring a factor of 16 or more output fl'om either ECR or
other sources that produce high charge states. To date the best known performance has
been flom the AECR-U source, which can produce about 0.8 puA of q=30 uranium

[Lyneis98] [LyneisOZ]. Since the present linac design requires two stages of stripping for

E N
ECR E 59" g:
g>[RF(j{Low p SRF .399 E

9.43 MeV/u Medium B SRF ‘3}
400 MeVlu

85.3 MeV/u @359
H H High 11 SRF j-[

B=0.81 B=0.6l B=0.49

Figure 3.5 Simplified layout of RIA driver linac under the acceleration scheme of
uranium. SRF signifies array of superconducting cavity structures.

 

 

 

 

103

uranium, only a few kilowatts of power would result from this type of source if only one
charge state is kept fl'om the ECR or after any stripper.

The entire linac is designed for transporting multi-q beams throughout, beginning
with extraction flom the ECR source. We shall not go into detail related to aspects of the
acceleration of multi-q beams, but instead cite the following references in which relevant
details are covered:

0 Bunching of a two-charge state (q=28 and 29) DC beam of uranium emitted flom
an ECR source followed by multi-q acceleration through an RFQ and low ,6 SRF
linac [OstroumovOOa].

o Acceleration of a five-charge state beam (q=73 to 77) of uranium through a
medium ,6 SRF linac after stripping at ~10 MeV/u [Ostroumov00b].

o Acceleration of a four-charge state beam (q=88 to 91) of uranium through a high
5 SRF linac after stripping at 85.3 MeV/u [Ostroumov99][Ostroumov00b].

The work on the topics listed above is ongoing and these references are not necessarily
the most recent. They do, however, offer a good overview of the overall plan for the RIA
driver linac.

The focus here will be on the aspects related to filtering out the unwanted charge
states and matching of multi-q beams to accelerating structures. These occur at the bends
shown in the diagram in Fig. 3.5. Since the beam has bunch structure after each of the
two foil strippers, there will be a requirement for rebunching and matching the
longitudinal phase space back into the next linac section. The bend sections must
incorporate achromat and isopath optical qualities in which the matrix approach is

valuable to the design. Since the higher order effects of the full system are of special

104

interest, it is necessary to apply higher order map calculations for the bunching and
focusing. Applying the newly developed COSY RF structures is useful for these
systems, and we shall show some examples of this.

The filtering of DC multi-q beams fl'om the ECR does not require bunch structure
enhancements until after q-state filtering. As we shall show later, the filtering requires a
simplified version of the systems required at the stripping sections and RF calculations
are not necessary. We shall focus on the required systems for filtering and rebunching
beams after the stripping stages and end the chapter with some detail about the stripping

process itself.

3.2.1 The conditions for an isopath
The term isopath is interpreted here as meaning equal path lengths for particles that

start at the same point but can vary in rigidity. According to (1.24) particles that differ in
rigidity fl'om the reference particle must have one or more of either 1%, 5,, or 5.. differing
from zero. At first we shall only regard the (X variable and then explain why it should
extend to the other two variables. We shall only consider reference particle motion in the
horizontal plane here, although the theory can be extended to motion beyond midplane
symmetry.

To develop an understanding of an isopath in terms of the transfer map, we should
first quantify the path difference. Suppose the reference particle travels from s.- to Sf sot
that it travels a distance, Lo, while some other particle of 5560 travel the distance, L, over
the same time interval of the motion. As long as each of their respective velocities

remain constant throughout then the path difference may be obtained from,

105

 

(1+yo) ‘ LO [ I (14.70))
AL=L— = ( - t,)=— 1+5 —— 1+——, (3.13)
L0 Cfl flo 70 (70(15‘70) L0 70

where we have applied Eq. (3.5) and the definition of 1 from Eq. (1.6). Since we shall
only consider the approximation in which l<<Lo, then we can drop the second term in the
large parenthesis and treat the results as exact. If we now substitute the relation for l in
terms of the initial position vector,

l=(l,x)x,. +(l,a)a,. +(l,5K )5,“ (3.14)
then the final form of (3.13) can be expressed as,

AL=--(-l—+—y°—){(l,x)xi +(l,a)a,. +[(l’6‘)-_L2—:l6‘}' (3.15)
r. 70(1+ro)
From this relation we see that in order to obtain an isopath (1.x) and (l,a) must vanish, and
that
(1,5 )=# (3.16)
" 70(1 + 70)

must also be satisfied. According to the symplectic relations (1.15g) and (1.15h), if (1.x)
and (l,a) vanish then (x,&) and (a,&) must vanish as well. Notice that the relations
(1.15e) and (1.151) also imply the same conditions. We infer that under these conditions,
having a system that is an isopath in 51' must imply that the system is also fully
achromatic.

The diagram in Fig. 3.6 sums up the situation discussed up to now. Notice that two
particles starting at the same position and direction will exit at the same point and
direction if the system is fully achromatic. In addition, if the system is also an isopath
then both particles would have traveled the same distance, L=Lo. We now consider the x

displacement away fl‘om the reference orbit,

106

x(s) = (x, 5,r )5“ , (3.17)
anywhere along s. To first order the momentary radius is given by R(l+hx), where the

variable h(s)=1/R(s) as defined in Chapter 1, and the derivative of the length, L, is given
by,

5:%;—=(l+hx). (3.18)

Furthermore, since for the reference particle x=0 over all s, the derivative of the path

difference must simplify to the relation,

%(L—L,)=%AL=hx. (3.19)

Finally, if we integrate (3.19) over the entire path and substitute x(s) from (3.17) into this

equation, then we obtain the integral relation,

AL = 5 K E, h(s)(x, 5 K )ds . (3.20)

To obtain an isopath system over the motion this integral must vanish. It should be noted

at this point that, since we have assumed that there is no change allowed in the velocity in

obtaining (3. 13), then there can be no acceleration in the direction of motion. Bending or

accelerating with electrostatic fields will not allow this condition to be satisfied;
therefore, the system should consist of only magnetic fields.

We can make some inferences about the type of magnetic field arrangement that will

Figure 3.6 Path of the
reference particle and some

arbitrary particle of length
Lo and L, respectively.

 

allow the integral in Eq. (3.20) to vanish and yiekl and an isopath. It is helpful to look at
the sets of paired plots in Fig. 3.7. They show two of the simplest schemes for obtaining
an achromat. The first achromat, shown by the diagram in Fig. 3.7 (a1), consists of a
mirror-symmetric dipole pair with an x-focus quadrupole at the center. The plot of its
dispersion function in Fig. 3.7 (a2) demonstrates how there is no way to make the integral
in Eq. (3.20) vanish. In this case because both the dispersion and h cannot chage sign
throughout.

The other type of achromat is shown in Fig. 3.7 (bl) and is constructed of four
dipoles, in which h>0 for the two outside ones and h<0 for the inner ones. Four dipole

systems have the possibility of having the integral vanish given the right combination of

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a1) * (b1) 3‘
3‘ 1 B1 82 B3 B4
A 2‘
’13 2* B1 Q B2 5 . W
v >. h>0 h>0
>" 1‘ 1. h<0 h<0
h>0 h>0
0 X(m) 0 X(m)
0'1'2'0'4'5'6 0 1 2‘3'4'
(a2) . (b2) 03 B1 B2 B3 B4
1* B1 B2 .

5o“ s(m) car 10 1 2 Hm)
5 0 - . Re 00 -
LJ 2 LJ 4 V \ /

1 h>0 h>0
, ._ ._. ._ __
Q . 0.34

Figure 3.7 Simplest possible pure magnetic achromatic systems. The possibility of
obtaining an isopath by the mirror synunetric system (a1) and four dipole system
(bl) is determinable from their respective dispersion functions, (a2) and (b2), as
described in the text.

108

bends. For simplicity we show the reference trajectory as coming in and exiting along
the same line; however, the actual arrangement would require an overall bend and/or shift
as we shall show later on.

As pointed in Chapter 2 the achromat feature is certainly necessary to avoid
horizontal dispersion fi'om the beam energy spread. Since the achromat feature depends
only on rigidity, then an achromat in 5. is also achromatic in 5,, and 5.; hence, (x,&),
(x, 5.), and (x,5,) vanish simultaneously. The same arguments that have been outlined for
(5: here also extend over to systems that are isopath with respect to variables 5, and 5...
Isopaths with respect to these two paramters have other unique features that may be
understood from the diagram in Fig. 3.6. Suppose that the two particles have the same
velocity but different charge. If the path lengths are equivalent, then so will the time of
arrival. This implies that the isopath condition with respect 5, will occur whenever (1,5,)
vanishes. The same applies when particles vary in mass, where an isopath exist for
(l,5..)=0. Keep in mind that systems having (1.50:0 are often referred to as being
isochronous in the sense that the particles are not relativistic (fl<<l). This condition does
not imply equal path length unless [3 z 1.

All three filtering sections of the RIA driver will certainly require that they be fully
achromatic. The first one, which filters DC beams flom the ECR, need only be of the
mirror-symmetric type in Fig. 3.7 (a1), since there is no need for bunch preservation.
The filter systems following the two strippers; however, will require rebunching and
longitudinal phase space matching for the sections of linac that follow them. Only
systems that have the isopath property, such as the system in Fig. 3.7 (bl), are

appropriate for this.

109

3. 2.2 Transporting and filtering of multiple q-charge state beams
As shown by Fig. 3.5 the RIA driver accelerator system requires two stages of charge

stripping and filtering of unwanted q-states for ions of 238U. The details about the
stripping and the effects that it has on the beam are left for the next section. In this
section we shall only focus on the design of systems that provide filtering as they match
and transport the accepted q—states to the section of linac that follows. The minimum
requirements of such a system are as follows:
0 The system provides dispersion region(s) for expunging unwanted q-states and
allows the accepted q-states to be transported with minimal emitttance growth.
0 It must rebunch the beam such that the longitudinal phase space characteristics
match with the next linac with for maximum acceptance.
0 At least at the rebuncher and exit regions, the system should also retain
achromatic character for minimum horizontal dispersion.
o Maintains isopath characteristics at the rebunching and exit regions.
The last requirement is presently the subject of some controversy. It is obvious that there
is the need for some control over the arrival time of bunches with different q; however, it
is not clear whether there may be some advantage gained at the proceeding linac by
manipulating the arrival time at the rebuncher. The study of the dynamics of multi-q
state beams along each section of the linac remains a topic that is open to exploring other
techniques. Such techniques may increase the acceptance when applying a shift in
energy and/or arrival time between q-states before the beam that enters the next linac
[Ostroumov00b]. In this study, we shall assume that the conditions required by the next

linac are such that each bunch arrives at the same time and energy. Thus, the phase of the

110

rebuncher shall be set such that A¢=0° according to Fig. 3.2 so that there is no shift in
time or energy between each q-state bunch.

To begin with, a detailed analysis of a 180° bend rebunching q-state filter at 85.3
MeV/u will be given, and then a description of a simpler alternative follows. After this
there will be a description of the solution for the bunching q—state filter after the 9.43
MeV/u stripper. That one will consist of a shift that leaves the linac sections offset, i.e. a
"dog-leg" illustrated in the diagram in Fig. 3.7.

3.2.2.1 The 180" bend rebunching q-state filter
The beam conditions for the system at 85.3 MeV/u have changed since initial stages

of the study. The filter has remained an 180° bend system since that allows the overall
length of the accelerator system to be shortened; however, it was initially intended to be a
system that accepts q-states 89, 90, and 91 of 238U. Since then there has been some
speculation as to whether stripping at this energy will actually yield maximum q-states
distribution at q=90 and whether the foil thickness predicted by the codes are accurate
(see next section). We shall stick with the initial predictions here, which state that the
beam shall have the longitudinal phase space parameters specified in Table 3.2 after
going through the stripper. The transverse phase space emittance has an area of 1.51:
mm-mr in both the x— and y—planes. The bounding ellipses have an upright position
and the beam has a circular shape with a diameter of 4 mm. The entire transport system
should deliver the beam to the entrance of the next linac with the same parameters and
having as few higher order effects as possible. Any changes in the design are expected to
be relatively small with little or no change in the dynamics of the system.

The initial design had assumed that a 350 MHz linac preceded the stripper section and

that the beam should be matched to a 700 MHz linac after the rebunching q-state filter.

111

Since a 350 MHz structure offers more acceptance with fewer longitudinal aberrations, it
is chosen for the rebunching cavities located at the midpoint of the system. We assumed
that there are two side-by-side two-gap structures located symmetrically about this
midpoint, which is labeled as 5 on the diagram shown in Fig. 3.8. The two-gap structures
have the parameters that are listed in Table 3.1. The parameters used for the elements
shown in the diagram of Fig. 3.8 are listed in Table 3.3. The dynamics that take place
can be broken down by the discussion that follows.

Without the RF cavities, the section between points 3 and 5 is that of a mirror-
symmetric dipole-pair achromat having 45° bends. The focus is telescopic from entrance
to exit in both the x— and y—planes. The horizontal focus is shown by the solid lines
representing rays of varying divergence. An x—focusing quad (Q5) sits at either side of
point 4 with enough room left between the two to allow the selection of dispersed charge
states with 88$qS92. Two trajectories of particles having q lower and higher than the
reference particle are shown by the dotted line. The largest separation between them
exists at point 4, where there is a charge dispersion of (x, 4)=-0.53 cm/%q. With this
dispersion the maximum separation between beams varying by one charge unit is ~6 mm
as shown by the x—y plot in Fig. 3.9. In the figure we have assumed Gaussian
distributions that will encompass 95.5% of the beam that lies within these phase space
boundaries. In this case, the boundaries will be circular with 3 mm radius and yields a
charge resolving power of qu200.

Since the higher q-states have a smaller radius at the dipoles, they end up traveling a

shorter path. The arrival time difference can be evaluated from (-At,4), which is

evaluated by dividing (1,4) by go to obtain units of time. A value of 4.68 ps/%q is

112

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RF 6
QS
Q5 Li?" I I
fly m\
-.LJQ-l I | I D
Q3
+Ar I | Me
[:1— fl/ §F§ 7
I
Q1
Q2
D1
D1
[2]—
m
in
Q1
02 —divue=me
| 1.5901“ l
111— (I) ”M Q —1:1

Figure 3.8 Diagram illustrating the 180° bend rebunching q-state filter. The system is
mirror symmetric about point 7. The first order longitudinal phase space ellipses are
shown for two q-states, one of which has charge qo and the other qo+Aq. The details of
the beam dynamics are described in the text.

obtained this way and yields that a q=91 ion arrives 5.2 ps faster (A¢=0.7° in terms of the
RF cycle) than a q=90 at the end of the mirror symmetric system. In order to offset this

effect, a chicane is placed before the 90° bend. Notice that the reference trajector enters

113

Table 3.3 Parameters used for the elements in Fig. 3.8.

Bend Bend
Dipole radius (cm) angle (°) strength (T)
D1 80.3 21.9 4.47
D2 71.8 45 5.0
Quad Strenght (T/m)
Q1 -24.09
Q2 24.09
Q3 18.70
Q4 -18.70
Q5 17.08
Q6 -17.49
Q7 17.83
Cavity parameters (see Appendix B)
frequency 350 MHz 40’ 0°
Max. Potential 949.2 kV
RI 2 cm RLRES 21.43 cm
R0 4 cm RRES 8.29 cm
RLDTl 12.86 cm RE ' 0.5 cm
RD 0.5 cm Cavity length 20.72 cm

0 3 cm aperture radius for all quadrupoles.
- Effective length of all quadrupoles is 30 cm.
0 All dipole pole gaps are taken to be 5 cm.

and exits along the same line, similar to the four dipole system depicted in Fig. 3.7 (bl).
By tuning this system such that the four D1 dipoles are at 4.47 T we can cancel out the
arrival time difference right after the second D2 dipole. The first order ellipses in the
longitudinal phase space are shown along the inner part of the diagram of Fig. 3.8 at each
point labeled from 1 through 9. Notice that the chicane causes the higher charge state
(+Aq) to arrive at position 3 later, in such a way that the bunches of varying q all arrive at
the RF cavities at the same time. We also estimate that since (-At,5()=0.58 ns/%K, a 215%

fluctuation in the phase of the cavity can impose a time spread of about 1.4 ps. This type

114

 

Figure 3.9 Plot of beam distribution for position 5 in Fig. 3.8.

of information is useful when considering the level of phase and frequency stability
needed at the RF structures.

We make note of the fact that the chicane is also a mirror symmetric achromat about
its center position (point 2). To first order, the multi-q beam is telescopically focused
from point 1 to 3 with magnifications of Mx=My=l; therefore, the transverse phase space
should remain the same as it started at point 1. The orientation of the longitudinal phase
space evolves as shown by the phase space plots. Notice that the orientation of the
longitudinal is the same as it started at point 1 due to the symmetry of the system about
point 5. In the second half of the sytem, the dynamics of the transverse plane evolve in a
similar way as in the first but in reverse.

3.2.2.2 Phase space calculations

The higher order effects on the multi-q beam are of major importance. Here, we will
demonstrate the effects of 2nd and 3rd aberrations from elements in the transport. It is
intuitively clear that particles of higher charge state will obtain stronger focusing in both
the transverse planes and the longitudinal plane (see next section). However, such effects
are not apparent until one goes to at least 2nd order as expected from magnetic fields of

mid-plane or double mid-plane symmetry [Wollnik87f]. Here, the focus will be

115

exclusively on the effects that are expected fl'om systems having only dipoles,
quadrupoles, and axially symmetric RF cavities.

The initial phase space distributions considered here are of the ellipse boundary-type
described in the first chapter. There are tluee boundary distributions of charge states
q=89, 90, and 91 that overlap each other initially. A similar approach was shown earlier
by Fig. 3.8 by a two q-state system in longitudinal phase space; however, only first order
effects were considered there. Here, we demonstrate the higher order effects on the phase
space, starting with the transverse phase space plots in Fig. 3.10. We show the phase
space that results right after the chicane at point 3 of Fig. 3.8. Only the results of the 2nd
order calculation are plotted since the 3rd order results are exactly the same for both x—
and y—phase space planes. First order results will show all ellipses overlapping with that
of the first order result for the 4:0 phase space. Effects from 1st or 3rd order
aberrations are not evident, since there are now axially symmetric devices. For the x—a
plot on the left the chromatic dispersion along the boundary for ellipses of different q is

attributed to the (x,x5k), (x,a4), (a,x5r), and (a,aék) terms. The aspect ratio between

lhb

 

(cw

 

 

 

Figure 3.10 Transverse phase space plots at point 3 in Fig. 3.8. Calculated with
COSY to 2nd order.

116

each q-ellipse also vary due the (x, 44) and (445:) terms. Similar effects are present

in thc y—b Pilaf»c Space from the (Myth). (yibdt). (bod). (b.1250. 055(4) and (b. 5151)
terms. As shown by the y—b plot at right, these effects are not nearly as strong as they
are in the x—a plane. There is somewhat of a different situation once the 90° bend
sections are included.

The plots in Fig. 3.11 are calculated for position 9, where the effects of the entire
system are included for the 2nd order (top pair of plots) and 3rd order (bottom pair of
plots) phase space calculations. From the y—b phase space plots it is evident that there
are now much stronger (y, 45.) and (b, 45k) effects present. These are attributed mostly
to the two 90" bend sections, and some minor contributions from each section of chicane.

The effects at 3rd order are attributed to achromatic focusing effects by the RF cavity.

Terms such as (y,yy4() and (x,xx5r) contribute to the blurring of the boundaries at each

 

 

 

 

 

 

Figure 3.11 Transverse phase space plots at point 3 in Fig. 3.8. Calculated with
COSY to 2nd order (top pair) and 3rd order (bottom pair).

117

respective phase space plane. An estimate is made of the emittance growth in the 3rd
order plots based on the amount of extra space that the particles occupy outside of the
boundary for the q=90 ellipse. There is ~50% growth in emittance in the x—a phase
space, while the y—b plane grows by ~30%. Most of the contributions to the emittance
growth in the x—a plane result from the elongation of the q=89 ellipse.

Finally, we show the effects on the longitudinal phase space plot in Fig. 3.12 as
calculated at the exit of the system at point 9 of Fig. 3.8. We have used the variable —At
instead of I for the horizontal axis. The maximum extent of the horizontal axis is also
shown in units of degrees of the RF phase and is 19°, which is equivalent to 150 ps in
terms of the period. Very weak achromatic effects are evident, and for the most part the
q-ellipses overlap with little relative shift at their boundaries. What does seem to be a
rather large effect is the 3rd order aberration, (I, &&&), of the buncher. The effect is due
to the large amount of drift space that is imposed by adding the chicane system on either

side. In order to reduce this effect we would need to switch to a lower harmonic

'5‘ (0.005)

 

-At
(150 ps)
[19°]

 

Figure 3.12 Longitudinal phase space ellipse at exit of 180° bend system.
Calculated to 3rd order with COSY. Boundary of all q-states overlap. Dashed

boundary ellipse contains 50% more area than the bounded phase space area.

118

frequency or come up with a more compact version of the isopath solution, as will be
shown later. A dotted line has been drawn which bounds the phase space in such a way
that a new emittance ellipse and orientation can be discerned. It sets the maximum limit
of the emittance growth under 3rd order conditions, which under these conditions the
boundary has ~50% more area than the area bounded by the initial ellipse. Notice that
the distortion on the ellipse makes the phase space seem as if it has a tilt much different
than the expected first order results. A more objective way to go about determining the
new ellipse characteristics for both the transverse and longitudinal phase spaces is to
evaluate the rms emittance and Twiss parameters by the statistical methods suggested at
the end of chapter 1. This requires a more realistic initial distribution, such as a Gaussian
or water-bag model, and a distribution of particles at each q-state according to the q-state
fractions described by the methods of the next section.

Based on these initial calculations it is certain that suppression of higher order
chromatic aberrations will be necessary. It was shown in the previous section that there
is sufficient dispersion at points 4 and 6 from Fig. 3.8 to separate the different charge
states. Any of these two regions is ideal for inserting higher order multipole fields, such
as hexapoles and octupoles, for suppressing the chromatic aberrations discussed above. It
was shown in the previous chapter how COSY can be used to find the best location and
field strengths necessary from the higher order multipoles to suppress geometrical
aberrations. Since the mirror-symmetric dipoles in the mass separator could not separate
the masses far enough in the multipole region, very little could be done about chromatic
aberrations there. In the case of the multi-q system here, there seems to be a negligible

amount of geometric aberrations in comparison with the chromatic aberrations. It is clear

119

that further studies with the q-filtering systems will be necessary to resolve such issues
for the final RIA design.

3. 2.2.3 Other isopath systems under consideration
The 180° isopath described above serves mainly as a proof of principle. Further

studies have allowed us to find other practical solutions in terms of shortening the path
length and reducing the number of necessary elements. A scheme that would replace the
function of the 180° isopath described earlier is shown in Fig. 3.13. The system is
essentially a hybrid between the mirror-symmetric 90° achromatic bend in Fig. 3.7(a1)
and the four-dipole chicane of Fig. 3.7(a2). The inside dipole bends beyond what an
ordinary chicane would in order to provide an overall bend by the entire system. With
the appropriate tune, the switch in polarity of the dispersion function will lead to a
vanishing of the integral in Eq. (3.20). Implementing this scheme into a 180° isopath
results in the system shown in Fig. 3.14. The first dipole (B1) bends the beam outward

by 10°, while the next one (B2) bends inward by 55° before the beam reaches the halfway

 

 

 

 

 

 

 

 

4 Q
‘1 32 B3
Bl B4
’8‘ 132 Q B3 _ ‘—
v 21
> an 0 F1 .11 \x 11" (m)
B B4 5 U5
‘ h>0 h>0 _ ‘
h<0 h<0
o - .
0 X (m) 5 '11

Figure 3.13 Diagram of an alternative scheme of obtaining an isopath system
with four dipoles and without the use of a chicane as in the system in Fig. 3.8
does (left). Plot of the dispersion function along the optic axis (right).

120

8': (m)
7 - rebunching
cavities

 

1 stripper foil exit

 

 

'2 I ' r ff' 7 r r

-2 -1 0 1 2
Figure 3.14 180° Bend scheme.

09-1
A:
a:
0)-

point of the achromat. The system is mirror symmetric about the midpoint of Q3 only if
the rebunching cavities are off. The entire system is mirror symmetric about the center
point of the cavity array even with the fields fi'om the RF cavity. We do not go into the
full details of this system, but instead refer to the literature where the work has been
reported [PortilloOl].

Lastly, we mention two other systems that are being considered for the design of the
RIA driver accelerator. The dog-leg system shown in Fig. 3.15 will be the rebunching q-
state filter following the first stripping stage of uranium at 9.43 MeV/u. It bends outward
at B; then back inward at B; and takes on an anti-symmetric geometry at the center point
of the expected superconducting RF (labeled as SRF) cavities. This system is unique in
that it is not truly a full achromat at the center point since (a,&) does not vanish. Since
the region of the cavity array is expected to be short (~1 m), then the horizontal

dispersion at positions away from the center should be negligible.

121

The advantage of the dog-leg scheme is that it leaves room for a branching out beam
line that makes use of beam that would otherwise be diverted to a dump. Instead it can
be diverted to a medium energy experimental area. It is expected that ~70% of the beam
would be accepted as a five charge state beam with reference charge qo~70. Since the q-
state fractional distribution is very symmetric, then about 15% of the unaccepted beam
can be diverted to the experiments and the other half is diverted to a beam dump. Notice
that we have not actually discussed how the unwanted multi-q states will be diverted
away from the accepted q-states. Future studies will need to incorporate septum magnets
for this purpose.

The last scheme, which is depicted in Fig. 3.16 is an option that had originally been
considered in an alternative design. It assumes that there are three stripping stages with
this one being the first one. It would strip a uranium beam of ~4 MeV/u at initial mean

charge state of 5328.5 up to 5:54. This system is actually not an isopath or full

achromat at the center and is designed such that there are minimal effects from the time
of arrival differences of each different q-bunch. It functions much like a chicane but with
rebunching at the center position. At present, this option has been ruled out due to
consideration of the extra beam losses in a relatively small cost section of the linac.

There are still issues that need to be resolved with the q-state filters. The higher order
effects for all these systems are still expected to be a problem if not corrected. Studies
are ongoing in this matter to finalize the designs. At this point, however, it has been
shown that the principles of the isopath transport can allow us to determine design

solutions that are practical for the RIA facility.

122

   

833 Q

4 10.11.12

 

 

'7 ‘ I I I . I . I I I * 1 r I .
0 2 4 6 8 10 12 (m)
Figure 3.15 Dog-leg scheme. The SRF stands location for superconducting
RF cavities.

16- (m)
14-

12-
' SRF

8? Q 520,, 03,933

6- Q B 3‘45 10.11.12 B4 Q

12 1 13.14

 

(In)

.4
-2'l'l‘lrl'1 l‘

-2 0 2 4 6 8'1'Or1'2'1'4'16

 

Figure 3.16 Chicane scheme.

123

3.3 Determining the distribution of q-states
3.3. 1 Importance of maximizing q

Maximizing the charge state of heavy ions in an accelerator system plays a critical
role in the design. One advantage of going to higher q is that lower field strengths for
bending and focusing of the beam are necessary. Note that the maximum energy that

may be bent by a magnet of rigidity, x," is given by the equation,

2 -12 2
KIA =-"—x—19—-gm2(%) , (3.21)

where the units are in MeV/u if the atomic mass unit, u=931.5 MeV/c2, and rigidity is in
Tesla-meter. K/A depends on the square of the charge of the particle. For example, at
q/A=90/238 and [..= 5 Teslax0.72 meter the maximum beam energy is 90 MeV/u. This
factor is especially important for cyclotrons, where x," is already set at the technological
limit of the magnet(s) and the only gain may come from increasing q.

Going to higher q will require fewer accelerating structures for attaining the final
energy of interest. In this case, the effect of increasing energy gain is linear with q/A.

This can be readily understood by expressing the energy gain per accelerating structure in

the form [OstroumovOOb],
AK =%ei,(/36) . T(,6, p6) - Lc(flG)-cos¢, (3.22)
Here, awn) is taken to be the average electric field over the effective gap length,

[4035), that is experienced by some particle traveling at a constant velocity, cflc. The

length £0660) is approximated to be the length of region in which the magnitude of the
field is above ~95% of maximum. In determining 1733090) it is assumed that the center

point between each gap is separated by some integral distance of the RF wavelength,

124

fig l(n+1/2) (see derivation of Eq. (3.11)), and also that the cavity phase is set to the
phase angle, ¢,,., where the ,BG particle gains maximum AK. The cosine term in the
equation accounts for the use of sinusoidal fields in the cavity at phase angle, 49,, relative
to 4),... q), is often referred to as the synchronous phase, since it usually is chosen such
that there is optimum stability in the oscillations of the bunch. Finally, the term, T(fl,fla),
is called the transit time factor and accounts for the fact that particles of velocity differing
from cflc will obtain less than optimal acceleration. Thus, it will always be less than
unity, except when [3%. Notice that, except for T(,6,,BG) and 49,, all terms may be
determined directly from the on-axis field, E,(s). The transit time factor may be
determined by evaluating AK at ¢,=O for every ,5. Although not demonstrated here, this
is an ideal application for the new COSY elements described in section 3.1.

Modeling arrays of structures in the form of Eq. (3.22) is the method used in
evaluating the final energy of the full driver linac. Accurate results will depend on how
well the process of stripping of electrons at foils is understood. As we shall demonstrate
below, the distribution of charge states depends on the incident energy of the particle. As
a result, there are many combinations of locations along the linac at which the beam can
be stripped, and each one will yield different charge distribution and final mean charge
state, a . Knowing the q-state distribution is also necessary to determine the
requirements for handling of the q-states to be accepted or dumped at q-filter systems of
the type described in the previous section.

The most challenging species of ions to strip and accelerate are those of 238U. At
any energy, it will yield the lowest q/A of any other stable ion and require energies above

~500 MeV/u to approach the fully stripped state (q/A =92/238=0.387). For the RIA

125

design it was necessary optimize for lowest cost by strategically determining the best
places along the linac to strip. As shown in Fig. 3.5 the two strippers are at ~10 and 85
MeV/u for the uranium beam. As we shall demonstrate below, the two energy regimes
are quite different and require their own respective analysis. A brief survey of the

available methods for determining q-states at each energy regime is given below.

3.3.2 The charge state evolution process

Before describing the techniques for evaluating q-states, it is instructive to look at the
underlying process. Consider that the particle may be at some charge state, q, and
additionally, in some electronic excited state. Neglecting nuclear excitation effects, the
collisions that the particle undergoes while traveling through some dense region of
material cause the following types of processes:

0 Excitation of the most loosely bound electron, or active electron, of a q-state ion
to higher excited states. In the energy level diagrams shown in Fig 3.17, this
would mean a jump to any state above the ground state within the left diagram.

0 Ionization from charge state q to q-I-l. The simplest case being that in which the
electron receives enough momentum from the collision to overcome the ground
state ionization potential, V;N=o(q) and go to the ground state of the q+l ion. The
arrow that goes from left to right between the diagrams illustrates this. The ion
can also undergo multiple excitations through multiple collisions before climbing
above the full V5, =()(q) potential

0 If a more tightly bound electron is extracted from an ion at q, then the q+l will be
left in an excited state (levels above ground on the right diagram). If there is

enough potential energy remaining, then the q+l ion may then de-excite by

126

emitting another electron (Auger process) and/or photon(s). This type of de-
excitation processes is associated with some lifetime and can occur even after
passing the material
0 The q-state ion can also capture an electron from the atoms/ions in the material
that it collides with. As such, the captured electron will go into either the ground
state of q-l or into one of its excited states. Radiative electron capture (REC)
occurs whenever there is an ejection of a photon as the electron is captured, which
is the inverse of the photoelectric effect. A non-radiative capture (NRC) occurs
when the electron is transferred from a bound state of a target electron to the
bound state of the active electron of the projectile.
Note that N is some integer that has been treated as a quantum number to represent the
state of the ion. Although a more detailed set of quantum numbers may be used, we
adopt this notation for simplification in which the ground state is at N=0 and all N>0 are

excited states.

we!)

 

{Mm

    

/
groundstate

Figure 3.17 Hypothetical quantum state level diagrams showing a simplified
transition from a q to q+l ionization state.

127

Every possible process should be associated with some cross section that specifies the
probability of occurrence with respect to the independent variable. Instead of using time
for the independent variable, it is usually more convenient to use the displacement, z, of
the particle along the direction of impact. The elapsed time may be extracted from the
average instantaneous velocity of the particles. Here, we shall interpret dq,N;q',N') as
the probability per unit depth of some ion at charge and quantum states, q and N,
respectively, going to states q' and N ' through either a collision event or spontaneous
process. We call Y4,” the fi'action of particles in the (q,N) state, such that the sum of the

fiactions over all possible states are conserved through

22”;qu =1. (3.23)
q

The derivative with respect to z of all the fi'actions form a system of linear coupled
differential equations in the form of [Betz72],

dY
A = 2 2b!“ -o'(q,N;q',N') — qu. -a(q'.N';q.N)]. (3.24)

dz anq'NalN'

As long as all the particles have the same mass then it is possible to separate them with

respect to their beam rigidity and measure the total yield at some q-state,

Yq =§YM =—JJi. (3.23)

The rigidity spectrometer cannot resolve excited states with a given q; thus, all N are

lumped together in flux, 14. Here, J is the total flux of all possible q-states. This equation
establishes the relationship between measurable and evaluated quantities.

In order to model the evolution of charge states, stopping powers, and kinematic

momentum transfer one must determine the cross sections for the most dominant

interactions. The cross sections will vary along the depth of the gas or solid as the

128

velocity of the projectile, vp, drops and the population of excited and ionization states
evolve. Due to the effect from screening of the nuclear charge many of these cross
sections tend to depend strongly on the velocity of the most loosely bound electron, or the
active electron, of the projectile. This velocity is approximately given by [Betz72],

v, = vanp’”3 , (3.24)
for an ion of charge state, q, that is in the ground state. Here, Z” is the atomic number of
the projectile, Z, is the atomic number of target atoms, and vo=ezlh=2.l88xx106 mls is
taken to be 1 a.u. (velocity in atomic units). This approximation is valid as long as (2,,-
q)23 and 2,236. In terms of the screening effect there are three distinct velocity regimes
described as follows:

0 In the low velocity regime (Z,,‘vp << v,“Z,) the electron capture cross section
dominates and a few interacting molecular states can be used to determine the
state of the ion.

0 In the high velocity regime (Z,.,’vp >> v,‘Z,) the excitation and ionization cross
sections dominate such that perturbative two-atomic state models determine the
state with sufficient accuracy.

0 In the intermediate velocity regime (Zp'vp ~ v.2.) the capture cross sections are
comparable in magnitude to the excitation and ionization cross sections. Thus,
developing models that reproduce experimental results is especially crucial in this
regime [Vemhet96].

Since there are a large number of possible excited states within each q-state, there are

necessarily quite a large number of cross sections to reference or evaluate. The number

of differential equations may also become so large that it becomes almost impossible to

129

achieve good accuracy and consistency. Thus, it is necessary to make as many
simplifications by grouping together any states that are similar and applying any available
scaling laws to reduce the amount of cross section data that needs to be stored and
referenced. We shall describe some codes that have been developed under this principle
and demonstrate some of the results that they yield.

Before going into details about the models, we should mention the concept of the
equilibrium charge state distribution. In order to demonstrate this concept results taken
from the ETACHA [Rozet96] code system and illustratedin Fig. 3.18. The plots are of
Yq in units of percent versus the depth into the target material, which we label as
thickness. The calculation assumes an ion beam of 238U incident at 24.1 MeV/u on
aluminum (Z;v,,/v,'Z,=0.09) at 2.7 g/cm3. The details of the calculations will be given
later, but for now we should like to point out the effects that occur after a sufficient
amount of interaction with the target has occurred. Notice that the fractions tend to level
off after a thickness of about 1 mg/cm2 as if to reach some equilibrium. The plot on the
top accounts for the average energy loss as the particle goes through the material. In this
particular case, the ion will lose about 1.4% of energy at 1 mg/cmz, and over 10 times
that amount by the time it gets to 10 mg/cmz. The energy loss has been evaluated
through numerical integration of the stopping powers, S(K)=dK/dz, which may be
obtained from the SRIM code system [SRIM98]. The plot at the bottom is for the yields
in which the cross sections are not corrected for the energy loss in the target. If the plot
were not logarithmic along the horizontal axis, then the two plots would look almost

identical.

130

ETACHA 24.1 MeV/u 238U on A]

m I I rvvvva r I II—IITI' I I I I ' I I I IIIII'

 

.0
O
O
corrected ..
.O. .00...
O .0000

 

  
 

 

 

 
   

 

 

 

' \ ’1' +81
I . O ’ ‘ t a . 4 . . . ‘ ‘ d —->—- w
. ‘ . ' -. "gong. .. _ ..
‘ n Yxp- MA A AAA-AuA-M-A” . u :— 79
6 .." I. - 78
. ’9‘) v ' \D ' v >-> , > . , s...» b v v . o
0. ,,.,.gs,;.,_,.;.;W.;.; :-:o:-:-:—-:-.; “'77
' Cl~~~ 76
50 . .m, .. m... . . mm, . . .-.m, + 75
. , +74
no correction + 73
40- . -—-72
—+— 71
_q__7o
‘ W " ' ' ----A---'$
. ’, 1............ 68
. ' . --I~- 67
\by « ‘ +66
. bio yyx .AuA-MndnmtAuA-M-A-m. “-0--- a
- . . . . ,A'
I ~ . z. I o .' "b ' . ‘
0 "t :- .t.-I?;i~3lt‘.°‘.é:§.:~:--;~-:':"" f ""' ""‘ f f i'f f
. . . ......, . ...fi... . ...mw . ..nm,
0.01 0.1 1 10 100
thickness (mg/cmz)

Figure 3.18 Charge state evolution of 238U on Al foil according to ETACHA. A
comparison is made between a calculation using cross sections that are corrected for
energy loss (top) and one without corrections (bottom).

Realistically, the corrected values will not completely reach some equilibrium
because the velocity of the particle is constantly being reduced by interactions with the
target material. The uncorrected plot, is merely a hypothetical process that is assumed to
occur in many measurements of equilibrium charge state distributions. For many

practical situations, such as this one, it is a reasonable approximation since for all Yq,N,

131

dYM
dz
after some depth. Whenever this type of equilibrium of fractions exist, or may be

-—) O , (3.24)

assumed to exist, we shall refer to the equilibrium distribution of the fractions, Yq, as
F (q). We shall first discuss some methods used to determine the equilibrium charge state
distribution through empirical parameterizations and then discuss other methods that

numerically integrate the set of ODEs in Eq. (3.24).

3.3.3 Empirical methods for determining F (q)
Target and particle systems in which q-state equilibrium regions have been observed

have been extensively studied and documented in the past [ReynoldsSS] [Baron88]
[Shima92] [Baron93] [Leon98]. With experimental data it has been possible to develop
methods of calculating equilibrium charge state distributions using semi-empirical
formulations. We shall provide the formulations prescribed by both [Baron93] and
[Leon98], since they are complementary to each other in terms of the range of species
that they cover. The following is a summary of the conditions upon which the
formulations are based on:

o The formulation in [Baron93] is based on extrapolations from experimental
results for ions species within 18 5 Z S 92 and energy range of 0.2 5 KM S 10.6
MeV/u.

o The formulation in [Leon98] is an extension of the work in [Baron93] with
additional data for species within 36 S Z,, S. 92 and energy range of 18 5 KM S 44
MeV/u.

The formulations are based on the assumption that the equilibrium q-state distribution is

symmetric about some value, a, , as long as the distribution does not extend to the region

132

of (Zp-q)<3. Although shell effects are observable in some studies [Shima92], they are
generally negligible for these approximations. For systems in which very few electrons
are left, the distribution is non-symmetric and the shell effects become pronounced
enough so that they may no longer be neglected, especially for helium like ions. The

form of the symmetric distribution is represented best by a Gaussian that is of the form,

 

F(q) = N1; exp[-(q - (7, )2 /2d2 )1, (3.25)
where by definition of F (q) we should have that the mean is given by,
(7. = Z q x F(q). (3.26)
q
and square of the deviation by,
.12 =Z(q-?I’.)2><F(q). (3.27)
q

The mean equilibrium charge is evaluated fi'om the semi-empirical formulation given by,

E. =Z,-f(Z,,.fl)-h(Z,.v,,.Z,)-g(Z,,). (3.28)
where the first of the three function carries the dependence on the projectile velocity,

Vp=flcv by

 

 

 

,
1 - lexp[— 832.2337x '6] [Baron93]
f (Z . ) = P (3.29)
" '6 1 —83.275x p
1 — exp 0.417 [Leon98]
L P
The formulation of Baron requires that
0.9 + 0.0769x (K / A) K/A > 1.3 MeV/u
1 = (3.30)
1 KM S 1.3 MeV/u

for best results. The target Z, dependence is embedded in the function,

133

[1 -5.21x10‘3(Z, -6)+9.56x10"(Z, -ti)2 -5.9x10"7(Z, —6)3] [Baron93]
h = V»

 

 

 

[(0.929 + 0.269 exp(-0. 160Z, )) + (0.022 - 0.249 exp(—0.3222, )) z m ] [Leon98]
P

(3.31)

Leon adopts the same form of the function, g, as given by Baron in the form,
g(z,,) =[1— exp(-12.905 + 0.2124(z,) -0.00122(z,,)2)] (3.32)

The form of the standard deviation is evaluated by the functions,
d _ 0.5.]; (1— ((7, IZP )‘fl [Baron93] 3 33)
1.1432J§,[0.07535+0.19(c7, Izp) 4.265407, /2,.)’] [Leon98]. '

The two models give about the same deviation within a few percent; however, they differ

in q; . The Baron model tends to give a higher a; by up to 10% at the low energy limit

for Zp~90. The difference diminishes with Zp until it is ~1% higher for Zp~30. The
difference becomes less at the higher energies until is only a fraction of a percent at ~40
MeV/u. At the high energy end, both models overestimate the mean charge state
compared to the available data in [Leon98] and agree with the data best at ~30 MeV/u.
The results given by the semi-empirical formulae are useful for approximating the
equilibrium charge state; however, they offer no information about the variation of the
distribution or of the thickness required for equilibrium. The studies done in [Leon98] do
not attempt to characterize the thickness, but merely mentions that for all their
measurements the thickness of the foils varied fi'om l jig/cm2 to several mg/cmz. On the
other hand, Baron offers the following formula for determining the equilibrium thickness

in jig/cm2 given that K/A is in units of MeV/u [Baron79]:

Dm[pg/cm2] =5.9+22.4(K/A) —1.13(K/A)2 (3.34)

134

In his equation, there is no attempt to put any Z, or Zp dependence. Much of the reason
for the lack of characterization with target thickness has to do with the erratic uniformity
found in most available thin foils. This applies especially for carbon foils, which are the
most commonly used. Measurements usually only provide an average value of the
thickness in the region of the beam; however, the actual thickness is very non-uniform
and it is not uncommon for it to deviate by as much as a by a factor of two from the
average value. This makes it difficult to correlate the thickness with the charge
distribution, and consequently, with the atomic numbers of projectile and target as well.
To obtain data that is associated with less uncertainty more stringent requirements will be
needed of the foil uniformity. The other alternative is to rely on accurately known cross
sections to extrapolate the thickness as explained below.

The work of Dmitriev focused on determining the equilibrium target thickness from
capture and stripping cross sections [Dmitriev82]. He takes the cross section of losing
the final electron before obtaining a fully stripped ion, a)(Zp-lzp), and the capture,
ac(Z,,, Z,- l), to the hydrogen-like ground state and applies it to the equation,

26. = Y,(0>-Y,(~)|exp[— 007mm +0.01. (3.35)
Here, the factor p=AflVA converts D into units of atoms/cmz, where A, is the atomic mass
of the target and NA is Avogadro's number. 174(0) and Yq(oc) are the initial and
equilibrium fractional yields for q= 5. Solving for 0(5) will, therefore, yield the
thickness necessary to obtain mean charge state a at equilibrium. The factor 8 is a

scaling factor that he determines should be about 0.01 to fit the data fi'om empirically
determined cross sections at energies below 100 MeV/u [Dmitriev79]. For energies

above this limit he adopts the cross sections from a Born approximation for electron loss

135

[Senashenko70]. The results for stripping of uranium are shown in the plot of Fig. 3.19,
along with those of the linear formula from Baron in (3.34).

Also plotted are some values that have either been reported as a measured quantity or
have been determined by charge evolution codes. The following explains the data points
labeled 1 to 3:

l. Experimentally measured equilibrium thickness values from [Scheidenberger98].

2. Experimentally measured equilibrium thickness values from [Leon98].

3. Evaluated using the codes GLOBAL and ETACHA.

There is little experimental work available in the literature about the stripping of uranium
at the region between ~40 and 100 MeV/u, thus we have resorted to the results of some
available codes that will be described later. The point at 45 MeV/u is at 21 mg/cm2 as

deduced from the results shown in Fig. 3.20 for an ETACHA calculation of U ions on a C

E

 

or; ------- [Baron79]
0 100 —G— [DHIIIIICVSZ]
E EB experimental data
,3, 10
:3
E 1
E
5:"
:g 0.1
i
0.01
1E-3
0.1 1 1 O 1 00 1000
KIA (MeV/u)

Figure 3.19 Equilibrium thickness for uranium beams according to
formulation by Baron and Dmitriev as explained in the text. Also shown are
values detennined as explained in the text.

136

foil. The point at 150 MeV/u is taken fi'om a calculation fi'om the code GLOBAL, which
indicates that equilibrium sets in at about 45 mg/cmz. ’

The formulation of Baron gets within 10% of the experimental value at 1.4 MeV/u
and within 30% for the data at 11.5 MeV/u. Beyond this point the approximation seems
to be no longer valid. The thickness evaluated by the model of Dmitriev seems to be in
agreement with the available codes for high energy, where the ions are fully stripped, but
fails to agree with the experimentally determined values at the lower energies. Neither of
the two models seems to predict the experimentally determined value at 24.1 MeV/u.
There is, however, some agreement at this point between the calculations shown in Fig.
3.18 (corrected) fi'om ETACHA for stripping with aluminum foils. The point at which
the fiaction for charge state 83 is a maximum occurs at ~4 mg/cm2 and it is also the point
at which there is maximum mean charge state. If this were to be interpreted as being the

point of equilibrium then there is good agreement with the experimentally determined

 

 

 

thickness [mg/cm2]
Figure 3.20 Evolution of charge states for K/A=45 MeV/u ions of U according to
the ETACHA code. The maximum mean charge occurs at ~21 mg/cm2 at which
point 2} =52.5 and 5 =0.69.

137

value of 4.7 mg/cm2 at point 2 in Fig. 3.19. For this measured value, [Leon98] provides
tabulated data of charge state distributions versus the thickness of the foil. Unfortunately,
this is not true of the values taken fi'om [Scheidenberger98]; however, the author assures

the reader that the results are obtained fiom measurements taken at equilibrium.

3.3.4 Codes available for determining charge state evolution
Up to this point we have not described codes that take the approach of solving the

form of Eq. (3.24) numerically. In principle, it is possible to apply this type of solution to
particles having energies within the range of the empirical formulations; however, the
number of electrons and excited states that the code must account for becomes too large
to be practical for heavy ions, such as uranium. Fortunately, the charge screening effects
cause most of the shell effects to be small enough that the empirical formulas can be
sealed with the available data to provide a reasonable prediction of the charge state
distribution.

At energies where the ions become almost fully stripped scaling laws tend to be less
reliable and it becomes necessary to account for quantum shell effects. Typically, the
excitation, ionization, NRC and REC capture cross sections need to be specified for each
Z, and Z, combination as a function of v, and the physical characteristics of the target,
such as the density. We provide a brief summary of two codes that can be used for
evaluating the evolution of charge states as a function of thickness.

3.3.4.1 ETACHA
The ETACHA code system has been tailored around experiments fi'om the GAN IL

cyclotron facility for ion energies ranging from 10 to 80 MeV/u for Ar, Ca, Fe, and Kr
(l8 2 Z S 36) on targets that range from Z,=1 to 54 [Rozet96]. The code solves the

system of first order differential equations through numerical methods by accounting for

138

up to 28 ebctrons (minimum charge state of q=Zp-28) distributed over the n=1, 2, and 3
subshells. It assumes that the first Born approximation, or plane-wave Born
approximation (PWBA) is valid as long as the ratio (v,/v,,)(Z,/Z,,)<0.35; i.e. Km>200
keV for p—>H collisions. As such, it evaluates excitation and ionization cross sections
from the PWBA using screened hydrogenic wave functions. For excitation and de-
excitation the code also allows interaction with the n=4 subshell by applying a l/n3
scaling law in evaluating the cross sections. In evaluating partial and total radiative
Auger decays, the code uses a method prescribed by [Larkins81].

For electron capture the code applies the Bethe-Salpeter formula [Bethe57], which
derives from an Eikonal approximation for evaluating NRC cross sections [Meyerhof85].
These approximations are best suited for the energy region in which (vva)(Z,/Z,,) stays
below ~l. Beyond this limit, the code allows the user to apply a more accurate method in
which NRC cross sections are evaluated through the continuum distorted wave (CDW)
approximation at the expense of more computing time [BelkiéS4]. The code also allows a
similar CDW approximation to be applied for evaluating the ionization cross sections
when (v,/v,,)(Z,/Z,,) is above unity.

In order to minimize the number of differential equations the calculation neglects any
spin orbital coupling effects. This allows the code to treat each electron within each of
the ls, 25, 2p, 3s, 3p, and 3d shells as being the same, and a scaling law is used to average
over the transition rates between the n=1 to 4 subshells and the orbital states within each
them.

To test the results from the ETACHA model we have compared the mean charge

states with those of tabulated experimental results for 24.1 MeV/u [Leon98]. The foils

139

 

 

 

 

 

 

 

 

90‘ —0—experirrent, [Leon98] ‘
1 calculation, EI‘ACHA‘
85- .
§BO-/ .
m
& J
WM
E 75* 9/ \O -
U 1
5 70- .
g .
65‘ d
. 1
60 . . .-. ... - a --e....
0.1 1 10
thickness (mg/cmz)

Figure 3.21 Mean charge states evaluated from charge state evolutions of 24.1
MeV/u 238U fiom experimentally measured values [Leon98] and from the code
ETACHA. The lines that intersect mark equilibrium thickness (0.7 mg/cm’) and
mean charge (5 =76.6) according to the analytical models.

are of aluminum and vary in thickness fi'om 0.2 to 13.5 mg/cmz. The results are plotted
in Fig. 3.21. The mean charge appears to be at a maximum at 5 =76.5 for 1.5 mg/cmz.
According to the results from the ETACHA calculation, at this same thickness 5 levels
off to a maximum of 82.3 (8% higher). The crossing lines indicate the thickness given by

Eq. (3.32) and the value of q, predicted by the empirical model of ref. [Leon98]. The

values are 0.7 mg/cm2 for the thickness and 5577.3. It seems that the ETACHA

calculations agree reasonably well with the data in terms of the onset of equilibrium but
not for the resulting mean charge state.

3.3.4.2 GLOBAL
Another code that evaluates the evolution of charge states is called GLOBAL

[Scheidenberger98]. It was developed after the ETACHA code and also takes into account

up to 28 electrons for the n=1 to 3 subshells. The scale factors that it uses are based on

140

data obtained for ions of energies ranging from 80 to 1000 MeV/u at the BEVALAC
accelerator at Lawrence Berkeley Laboratory [Alons082] and the SIS synchrotron at GSI
in Darmstadt. The cross sections used for excitation and ionization are similar to the
PWBA used in the ETACHA code but has in addition a relativistic factor included. The
code is recommended for use with projectiles of Xe to U (54 2 Z1p S 92) impinging on
solid or gaseous targets of Be to U (4 2 Z, S 92). The work of Sheidenberger and others
makes comparisons between compiled experimental results and the predictions of the
ETACHA code and finds that there is reasonable agreement up to ~30 MeV/u. They
concede that the results of GLOBAL are not very accurate below ~100 MeV/u and that
there is no technique available to make good predictions in the range of 30 MeV/u to 100
MeV/u. Unfortunately, this is in the regime at which the second stripper for the RIA
driver linac intends to strip. We shall discuss some of the results predicted by the codes

at both the first and second stripper and how they apply to RIA.

3.3.5 Estimating q-state distributions for the RIA driver
Some of the results fiom the ETACHA and GLOBAL codes that are used as

examples in the discussions leading up to here are actually results relevant to the RIA
driver linac. In particular, the results plotted in Figs. 3.18 and 3.20 (corrected) deserve
further discussion. Notice in going fi'om 24.1 MeV/u to 45 MeV/u that the region that
could be considered equilibrium becomes narrower. In other words, the flat region that
appeared before is no longer there, and it is not obvious that any such equilibrium
condition even exists. It seems that right at the point at which the ion begins to approach
equilibrium it begins to slow down considerably to the point where a reverse process

begins to take effect, as shown by the drop in charge state. Clearly, the models indicate

141

that the stopping powers become a very important factor as the ion approaches the H- and

He-like states. For the 45 MeV/u case, the highest mean charge, in , and lowest standard

deviation in charge occur at about 21 mg/cmz. In arriving at the equilibrium thickness

values plotted in Fig. 3.19 and labebd as 3, we assumed that a, can. Furthermore,

calculations from GLOBAL at 45 MeV/u yield a similar thickness and mean charge
within a fraction of a percent of those of ETACHA. This should not be surprising, since
the cross sections of GLOBAL are similar to those of ETACHA when relativistic effects
become negligible.

At the high energy end of the spectrum, GLOBAL seems to indicate that the
equilibrium-type behavior becomes pronounced once again. This can be seen from the
plot shown in Fig. 3.22, where the evolution has been plotted for 150 MeV/u 238U ions
on carbon. This result had been used for choosing the other equilibrium thickness labeled
as 3 in the plot of Fig. 3.19. Evidently, the equilibrium behavior seems to be less

prominent after an energy of ~40 MeV/u and reappears gradually after ~100 MeV/u. The

 

  

 

 

0.9 .fijfilnr

o 8 150 MeV/u 238U on Carbon

- 1
+”
0.7 ‘ ‘ .--o..- 91
g 0.6 ‘ .. *m
'3 05 ‘ ‘ +89
3 ' j —<>—88
2.1:. 0.4 . : e +87
. - --->t--86
0.2 1 i) . +34
01;”! ’ ”"ILIIILIIM‘: ""”“°33
00 ' Atéz .ar-l-I . I‘m-z::::;::;.;;.....,..,t
...To . .e...ri60 . .
. 2
tluckness (mg/cm )

Figure 3.22 Evolution of charge states at the high energy range of uranium on
carbon as calculated by GLOBAL.

142

work of Gould and others [Gould84] at the Bevalac reports the equilibrium at the higher
energy regime for stripping of uranium. There they have stripped at energies as high as
962 MeV/u where they obtained 85% of the beam stripped to q=92 starting from q=68.
The minimum foil thickness needed to reach the plateau region of equilibrium was 150
mg/cm2 with Cu and 85 mg/cm2 with Ta. At 437 MeV/u they needed 90 mg/cm2 Cu to
obtain 50% of the beam in the q=92 state. The same group later reported the results from
200 MeV/u uranium ions on foils of Mylar (236.6), aluminum, copper, and silver
[Gould85]. Equilibriums are observed there as well, unfortunately, they do not report the

minimum thickness needed from each material The authors make some attempt to

characterize the highest mean charge, in , possible using different Z. from scaling done
with capture and ionization cross sections. They claim that higher a, are obtained with

high Z, at the high energies, and that there is a gradual trend so that at the lower energies

lower Z, give higher 5.. . Their lower energies imply <200 MeV/u, which would apply

more to our studies at >80 MeV/u. From stripping at energies of ~500 MeV/u they find
that some intermediate Z, gives the higher in .

Clearly, there is experimental evidence of an extended plateau that indicates there is
some equilibrium condition reached at the lower and higher energies. The plateau region
is convenient for selecting a minimum thickness for the foil. Unfortunately, the codes
have not shown this type of plateau region to exist for the intermediate energy region, and
in some cases even the low energy region. The latest design of the RIA driver linac
requires that