Optics of Charged Particles

310 Pages·2021·15.85 MB·English

by  Hermann Wollnik

O C
PTICS OF HARGED
P
ARTICLES
O C
PTICS OF HARGED
P
ARTICLES
Second Edition
HERMANN WOLLNIK
New Mexico State University, Las Cruces,
NM, United States
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Contents
Forewordtosecondedition xi
Foreword xiii
1. Gaussian optics and transfer matrices 1
1.1 Themethodoftransfermatrices 1
1.1.1 Thedescriptionofstraightraysinopticalsystems 1
1.1.2 Propertiesofthinlenses 3
1.2 Anopticalsystemthatcontainsonethinlens 4
1.2.1 Theformationofarealimageorofavirtualimage 6
1.3 Generalopticalsystems 8
1.3.1 Thesignificanceofthedisappearanceofelementsof
transfermatrices 10
1.4 Transfermatricesoflensmultiplets 11
1.4.1 Thetransfermatrixofadoubletoftwothinlenses 12
1.5 Roundlensesforchargedparticles 18
References 19
2. General relations for the motionof charged particles in
electromagnetic fields 21
2.1 Energy,velocity,andmassofacceleratedchargedparticles 21
2.2 Forcesonchargedparticlesinelectromagneticfields 22
2.2.1 Themagneticrigidity 25
2.2.2 Theelectrostaticrigidity 26
2.3 Thedescriptionofabundleofparticlesofdifferentkinetic
energiesandmasses 28
2.4 Therefractiveindexoftheelectromagneticfield 29
2.5 TheEuler(cid:1)Lagrangeequations 32
Appendix 33
Hamilton’svariationalprincipleandtheLagrangeequations 33
Themotionofchargedparticlesintime-independent
magneticandelectrostaticfields 35
References 37
3. Quadrupolelenses 39
3.1 Theelectricandmagneticfieldsinquadrupolelenses 40
3.1.1 Fieldsinelectricquadrupolelenses 41
v
vi Contents
3.1.2 Fieldsinmagneticquadrupolelenses 42
3.2 Particletrajectoriesinquadrupolelenses 43
3.2.1 Theequationsofmotion 43
3.2.2 Focusingpropertiesofsinglequadrupolelenses 46
3.3 Thedesignofquadrupolemultiplets 49
3.3.1 Quadrupoledoublets 50
3.3.2 Quadrupolemultiplets 56
3.4 Thin-lensapproximationsforquadrupolemultiplets 59
3.4.1 Doublypoint-to-parallelfocusingthin-lensquadrupole
multiplets 60
3.4.2 Stigmaticfocusingthin-lensquadrupolemultiplets 63
3.4.3 ThinlensquadrupolemultipletsforwhichðxjaÞand
ðbjbÞvanishsimultaneously 65
3.4.4 A“beamrotator” 65
References 66
4. Sector fields 67
4.1 Homogeneousmagneticsectorfields 68
4.1.1 Particletrajectoriesinhomogeneousmagneticsectorfields 68
4.1.2 Focusinganddispersingpropertiesofhomogeneousmagnetic
sectorfields 72
4.1.3 Obliqueentranceandexitofparticlebeamsinsectormagnets 81
4.2 Inhomogeneousmagneticsectorfieldsformedbyinclinedplanarpole
faces(wedgemagnets) 86
4.2.1 Particletrajectoriesinwedgemagnets 87
4.2.2 Focusinganddispersingpropertiesofwedgemagnets 90
4.3 Radiallyinhomogeneoussectorfieldsformedbyconicalpolefacesor
toroidalelectrodes 92
4.3.1 Theequationsofmotioninradiallyinhomogeneoussectorfields 93
4.3.2 Particletrajectoriesinradiallyinhomogeneousmagneticor
electrostaticsectorfields 98
4.3.3 Focusinganddispersingpropertiesofradiallyinhomogeneous
sectorfields 100
4.3.4 Examplesofradiallyinhomogeneoussectorfields 104
4.4 Particleflighttimesinradiallyinhomogeneoussectorfields,
quadrupoles,andfield-freeregions 106
Appendix 109
Paraxialtrajectoriesinwedgemagnets 109
References 112
Contents vii
5. Chargedparticlebeams inphase space 113
5.1 Liouville’stheoremandfirst-ordertransfermatrices 114
5.2 Phase-spaceareasofparticlebeamspassingthroughopticalsystems 115
5.2.1 Phase-spaceareasofparticlebeamsinfield-freeregions 116
5.2.2 Phase-spaceareasofparticlebeamsinimage-formingsystems 117
5.2.3 Thevirtualthinobjectlens 118
5.3 Beamenvelopes 120
5.3.1 Envelopesofbeamswithparallelogram-likephase-spaceareas 122
5.3.2 Envelopesofbeamswithoctagon-likephase-spaceareas 123
5.3.3 Envelopesofbeamswithellipticalphase-spaceareas 123
5.4 Positionsandsizesofenvelopeminima 129
5.4.1 Imagesandpupils 129
5.4.2 Beamwaists 130
5.4.3 Pupilsandwaistsforpoint-to-parallelfocusingsystems 131
5.5 Minimalbeamenvelopesatpostulatedlocations 131
5.6 Liouville’stheoremanditsapplicationtowide-anglebeams 134
5.6.1 Abbe’s“SineLaw”foratwo-dimensionalphasespace 135
5.6.2 Thecurrentdensity 136
5.7 Beamswithspacecharge 137
5.7.1 Thecompensationofspacechargeinparticlebeams 138
5.7.2 Thecalculationoftransporteffectsinconstantdiameter
ionbeams 139
5.7.3 Thedesignofbeamlinesforhigh-intensitybeams 141
References 143
6. Particle beams inperiodic structures 145
6.1 Single-particletrajectoriesandbeamenvelopes 146
6.1.1 Systemswithpostulatedopticalproperties 149
6.1.2 Opticalsystemsthathaveidenticalentranceandexitbeams 151
6.2 Ringsofunitcells 158
6.2.1 Lateralbeamdeviations 158
6.2.2 Longitudinalbeamdeviations 159
References 160
7. Fringe fields 161
7.1 Particletrajectoriesinthefringefieldsofdipolemagnets 161
7.1.1 Positionsofeffectivefieldboundariesindipolemagnets 163
7.1.2 Fringe-fieldshuntsfordipolemagnets 165
viii Contents
7.1.3 Fringe-fieldshuntsforhomogeneousandinhomogeneous
dipolemagnets 169
7.1.4 y-Focusinginthefringefieldsofdipolemagnets 169
7.1.5 Transfermatricesofrealisticmagneticsectorfields 174
7.2 Particletrajectoriesinfringefieldsofelectrostaticdeflectors 174
7.2.1 Electrostaticdeflectorsbiasedsymmetricallytoground 175
7.2.2 Electrostaticdeflectorsbiasedasymmetricallytoground 176
7.2.3 Transfermatricesofrealisticelectrostaticsectorfields 178
7.3 Particletrajectoriesinthefringefieldsofquadrupolelenses 179
References 182
8. Image aberrations 183
8.1 Systematicsofimageaberrations 183
8.2 Originofimageaberrations 190
8.2.1 Geometricaberrations 191
8.2.2 Chromaticaberrations 196
8.3 RelationsbetweencoefficientsofEq.(8.2)duetotheconditionof
symplecticity 201
8.3.1 Canonicaltransformationswithtimebeingtheindependentvariable 201
8.3.2 Canonicaltransformationswithpositionbeingthe
independentvariable 202
8.3.3 Theconditionofsymplecticity 204
8.4 Imageaberrationsofnthorder 211
8.4.1 Overallimageaberrations 213
8.4.2 Theaberrationdrivingtermsofallorders 217
8.4.3 Imageaberrationsduetofringefields 217
Appendix 222
Coefficientsofimageaberrationsofnthorder 222
Equationsofmotionintime-independentfields 223
Explicitparticletrajectoriesinradiallyinhomogeneouselectromagnetic
sectorfields 224
References 228
9. Design of particle spectrometers and beamguide lines 229
9.1 Single-sectormagneticspectrometers 229
9.1.1 Ionbeamsfrompoint-likeandslit-likeionsources 232
9.1.2 Asectormagnetionanalyzerassistedbyonequadrupolelens 233
9.1.3 Aphase-spaceadapterforionbeamsinsectormagnets 234
9.2 Aqualityfactorforparticlespectrometers 235
9.2.1 Arigidity-orQΔ-value 236
Contents ix
9.2.2 AnenergyQ -valueandamassQ -value 240
K m
9.2.3 AQ-valueforNcascadedspectrometers 241
9.2.4 Achromaticsystems 242
9.3 Time-of-flightparticlespectrometers 245
9.3.1 AQ-valuefortime-of-flightmassanalyzers 247
t
9.3.2 Isochronousopticalsystems 248
9.3.3 Angleandenergy-focusingisochronousopticalsystems 249
9.4 Thealignmentofanopticalsystemandthecorrectionofits
aberrations 251
9.4.1 Thecorrectionofthesecond-orderapertureaberrationsin
magneticsystems 253
9.4.2 Thecorrectionofimageaberrationsinelectrostaticsystems 263
References 265
10.Time-of-flight mass spectrographs 267
10.1 Time-of-flightmassspectrographsthatusesectorfields 269
10.1.1 Time-of-flightmassspectrographsthatusemagnetic
sectorfields 269
10.1.2 Acceleratorstorageringsusedastime-of-flightmass
spectrographs 270
10.1.3 Determiningthepassagetimeforhigh-energyions 271
10.2 Time-of-flightmassspectrographsthatuseelectrostaticsectorfields 272
10.3 Low-energyionbeamsintime-of-flightmassspectrographs 275
10.3.1 Ioncoolinginaquadrupoleiontrap 275
10.3.2 Coolingionsinandextractingthemoutofaflattrap 277
10.3.3 RF-carpetstointroduceionsintocoolers 277
10.4 Acceleratingandbunchingofionsinpulsedelectricfields 278
10.4.1 Acceleratingandbunchingbyasinglepulsedfield 279
10.4.2 Ionbunchingtoadesiredposition 281
10.4.3 Delayedionextraction 283
10.5 Energy-isochronoustime-of-flightmassspectrographsthatuse
ionmirrors 285
10.5.1 Energy-isochronoustime-of-flightmassspectrographsthat
usegriddedionmirrors 285
10.5.2 Energy-isochronoustime-of-flightmassspectrographsthat
usegrid-freeionmirrors 287
10.5.3 Energy-isochronoustime-of-flightmassspectrographsthat
useseveralionmirrors 287
10.6 Amultireflectiontime-of-flightmassspectrograph 288
10.6.1 MovingionsintoandextractingthemoutofaMRTOF-MS 290
x Contents
10.6.2 Determiningthenumberoflapsthationshaveperformed
inamultireflectiontime-of-flightmassspectrograph 291
10.6.3 ThedesignofaMRTOF-MSforprecisemassdeterminations 292
10.6.4 Achievablemassresolvingpowers 293
10.6.5 ORBITRAP:asmallhighfrequencymultireflectiontime-of-flight
massspectrograph 296
References 297
Index 301
Foreword to second edition
The second edition of Optics of Charged Particles follows the first edition
after more than 30 years and attempts to outline again how charged parti-
cles move in magnetic and electric fields. In this second edition, newly
arisen questions are addressed as, for instance, those of new time-of-flight
measurements, and old explanations are improved in simplicity and com-
pleteness describing how charged particles move through the main and
the fringe fields of optical systems and through individual optical elements.
Also, it uses the same type of formulation and nomenclature throughout
the book.
As with the first edition, the second edition does not require the
reader to have any special preknowledge of charged particle optics.
However, it requires that the reader has some basic understanding of the
physics and mathematics equivalent to an undergraduate education. The
intention of the second edition is also not to lead the reader to individual
mountain tops of scientific findings, but rather to a high plateau of under-
standing from where the mountain tops can be seen and reached by his or
her own strength and skill.
I have tried to supply a comprehensive set of references, which include
the earliest publications as well as the most important ones. However, not
all possible references are listed, as this would have exceeded the available
space.
At this point, I would like to acknowledge the many fruitful discussions
and insights about the object of charged particle optics with my colleagues
M. Berz, K. Brown, S. Dodonov, K. Halbach, A. Kalimov, H. Matsuda,
T. Matsuo, S. Schepunov, P. Schury, D. Vieira, and M. Wada. I also
extend my thanks to my wife Annette who constantly supported my efforts
over so many years.
Santa Fe, 03-05-2021
Hermann Wollnik
xi