Analytical, approximate-analytical and numerical methods in the design of energy analyzers

212 Pages·2015·14.35 MB·English

by  Gurov

EDITOR-IN-CHIEF
Peter W. Hawkes
CEMES-CNRS
Toulouse, France
Coverphotocredit:
VictorS.Gurov,ArmanO.Saulebekov,AndreyA.Trubitsyn
Analytical,Approximate-AnalyticalandNumericalMethodsintheDesignofEnergyAnalyzers
AdvancesinImagingandElectronPhysics(2015)192,pp.1–14
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FOREWORD
Themainapplicationofanenergyanalyzerofchargedparticles,thedesign
ofwhichconstitutesthecontentofthisbook,iselectronspectroscopy.The
greatvarietyofthetasksandmethodsofelectronspectroscopyisthereason
forthemultiplerequirementsthataregenerallyimposedonenergyanalyzer
design.Nevertheless,twoprincipalparameterscanbedistinguishedtochar-
acterizeanenergyanalyzerasanelectronspectrometrydevice:namely,the
orderofangularfocusingandtheenergydispersion.Thefirstoftheseparam-
eters determines the sensitivity of the energy analyzer, while the second
determines its energy resolution. The nature of these parameters is self-
contradictory,whichmanifestsitselfinthefactthattheattemptsofimprov-
ingoneofthemcommonlyleadtodeteriorationoftheother.Thus,thekey
tosolvingtheproblemofoptimaldesignofenergyanalyzersliesintheways
toreducetheextentofthisconflict.Inparticular,apartialresolutionofthis
physicalcontradictionmaybeachievedbyusingtheelectrostaticfieldswith
low spherical aberration, or, alternatively, by creating the conditions for
high-order angular and spatial focusing.
Inthisbook,wediscussthreedifferentapproachestotheanalysisofthe
focusinganddispersiveelectrostaticfieldsthatareusedinenergyanalyzers.
These comprise the analytical methods, which allow researchers to find a
solution in terms of elementary mathematical functions in some cases; the
approximate-analytical methods, when a solution is sought in the form of
infiniteseries;and,finally,thenumericalmethodsthatemployallthepower
of the modern methods of computational mathematics and provide a solu-
tionintheformofafunctiondefinedonameshwitharestrictednumberof
nodes.Inthiscontext,theauthorswouldliketostatetheirconfidencethat
successinthedesignofnewtypesofenergyanalyzersundoubtedlyrequiresa
reasonable combination of all three of the approaches.
Theoriginalstudiesoutlinedinthisbooksummarizetheresultsofmore
than 20 years of workof each of theauthors in thefield of energyanalysis,
friendly discussions on all areas of electron optics, and independent exami-
nation of the devices designed. The book can be useful for students of
chargedparticleoptics,scientistsandengineersinvolvedinthedevelopment
of the new advanced energy analyzers, and users of electron spectroscopy
devices who want to keep abreast of current trends in this area.
vii
viii Foreword
All the chapters are the result of collective work; however, we should
distinguishabasiccontributionofeachoftheauthorsinwritingtherelevant
chapters.Chapter1hasageneralintroductorycharacter.Chapter2,which
presents analytical methods for the design of certain types of energy ana-
lyzers,hasbeenbasicallywrittenbyProfessor V.S.Gurov.Themain con-
tribution to Chapter 3, dedicated to the approximate-analytical methods,
was made by Professor A. O. Saulebekov. Finally, the content of
Chapter 4, which outlines the numerical methods as applied to designing
the energy analyzers, is primarily the work of Professor A. A. Trubitsyn.
Inconclusion,theauthorsexpresstheirsincererespectforanddeepgrat-
itudetotheircolleaguesandteacherswho,giventheleveloftheirexcellent
work and general contribution to science, can be rightly regarded as the
“aces” of electron and ion optics–namely, V. V. Zashkvara, S. Ya. Yavor,
E. P. Sheretov, K. Sh. Chokin, B. U. Ashimbayeva, and V. A. Gorelik.
Without their kind attention and fruitful contacts with the authors in the
past and present, this monograph would not have been produced. Special
thanksareextendedtoDr.M.A.Monastyrskiy,whotranslatedthistextinto
Englishandmadeanumberofexceedinglyusefulprofessionalcommentson
the form and content of virtually all its sections.
We also highly appreciate the financial support of the Russian Science
Foundation (grant No. 15-19-00132) in the creation of this publication.
VICTOR S. GUROV
ARMAN O. SAULEBEKOV
ANDREY A. TRUBITSYN
PREFACE
The theme of the present volume is the design of energy analysers for
charged particles and in particular, the various approaches that have been
developedforthetask:analyticalmethods,approximateanalyticalmethods
for more realistic field models and numerical methods.
Thebookopenswithasurveyofthetraditionaltypesofenergyanalyser
and of more advanced models. This sets the scene for the three families of
methodsemployedinthedesignoftheseinstruments.Chapter2coversthe
analytical methods based on cylindrical and hyperbolic field models. This
isfollowedbyachapteronmorerealisticfieldmodelsforwhichthemath-
ematical analysis requires certain approximations. Several hexapole–
cylindrical analysers are studied in this way.
Useful though model fields are to establish the general behaviour of an
analyser and its dependence on the various parameters present, an exact
simulation requires numerical methods and these are the subject of
Chapter4.Suchmethodsaredescribedinconsiderabledetailandthechapter
ends with several examples of simulations of different types of analysers.
Iamverygratefultotheauthorsforpreparingthisclearandfullaccount
ofthesubject.TheoriginalRussiantexthasbeentranslatedbyDrMikhail
Monastyrskiy,whosenameisalreadyfamiliartoreadersoftheseAdvancesand
who has published many papers on this and related subjects. I am sure that
readerswilljoinmeinthankinghimforundertakingthischallengingtask.
PETER W. HAWKES
ix
FUTURE CONTRIBUTIONS
S.Ando
Gradientoperatorsandedgeandcornerdetection
J.Angulo
Mathematicalmorphologyforcomplexandquaternion-valuedimages
D.Batchelor
Softx-raymicroscopy
E.BayroCorrochano
Quaternionwavelettransforms
C.Beeli
Structureandmicroscopyofquasicrystals
C.BobischandR.Mo€ller
Ballisticelectronmicroscopy
F.Bociort
Saddle-pointmethodsinlensdesign
K.Bredies
Diffusiontensorimaging
A.Broers
Aretrospective
N.ChandraandR.Ghosh
Quantumentanglementinelectronoptics
A.CornejoRodriguezandF.GranadosAgustin
Ronchigramquantification
L.D.DuffyandA.Dragt(Vol.193)
Eigen-emittance
J.Elorza
Fuzzyoperators
R.G.Forbes
Liquidmetalionsources
P.L.GaiandE.D.Boyes
Aberration-correctedenvironmentalmicroscopy
M.Haschke
Micro-XRFexcitationinthescanningelectronmicroscope
R.HerringandB.McMorran
Electronvortexbeams
M.S.Isaacson
EarlySTEMdevelopment
xi
xii FutureContributions
K.Ishizuka
Contrasttransferandcrystalimages
K.Jensen,D.ShifflerandJ.Luginsland
Physicsoffieldemissioncoldcathodes
M.Jourlin(vol.194)
Logarithmicimageprocessing,theLIPmodel.Theoryandapplications
U.Kaiser
Thesub-A˚ngstro€mlow-voltageelectronmicroscopeproject(SALVE)
C.T.Koch
In-lineelectronholography
O.L.Krivanek
Aberration-correctedSTEM
M.Kroupa
TheTimepixdetectoranditsapplications
B.Lencova´
Moderndevelopmentsinelectronopticalcalculations
H.Lichte
Developmentsinelectronholography
M.Matsuya
CalculationofaberrationcoefficientsusingLiealgebra
J.A.Monsoriu
Fractalzoneplates
L.Muray
Miniatureelectronopticsandapplications
M.A.O’Keefe
Electronimagesimulation
V.Ortalan
Ultrafastelectronmicroscopy
D.Paganin,T.GureyevandK.Pavlov
Intensity-linearmethodsininverseimaging
N.PapamarkosandA.Kesidis
TheinverseHoughtransform
Q.RamasseandR.Brydson
TheSuperSTEMlaboratory
B.RiegerandA.J.Koster
Imageformationincryo-electronmicroscopy
P.RoccaandM.Donelli
Imagingofdielectricobjects
J.Rodenburg
Lenslessimaging
FutureContributions xiii
J.Rouse,H.-n.LiuandE.Munro
Theroleofdifferentialalgebrainelectronoptics
J.Sa´nchez
Fishervectorencodingfortheclassificationofnaturalimages
P.Santi
Lightsheetfluorescencemicroscopy
R.Shimizu,T.IkutaandY.Takai
Defocusimagemodulationprocessinginrealtime
T.Soma
Focus-deflectionsystemsandtheirapplications
I.F.Spivak-Lavrov(Vol.193)
Analyticalmethodsofcalculationandsimulationofnewschemesofstaticandtime-of-flight
massspectrometers
J.Valde´s
RecentdevelopmentsconcerningtheSyste`meInternational(SI)
CHAPTER ONE
Energy Analysis of Charged
Particle Flows
Contents
1. BasicParameters 1
2. MainTypesofEnergyAnalyzers 2
3. AdvancedEnergyAnalyzers 8
Energyanalyzersservetodetectchargedparticleswithprescribedcharacter-
isticenergies,or,inotherwords,toisolatechargedparticles,theenergiesof
which fall in a narrow energy interval ΔE.
1. BASIC PARAMETERS
The main consumer parameter of energy analyzers is energy resolu-
tion. Two definitions of energy resolution are used in spectroscopy (see,
for example, Seah & Briggs, 1992). The first definition is associated with
the absolute energy resolution ΔE, representing the entire energy range
within which the particles reach the collector. Such energy resolution is
called the basic resolution. In practice, the resolution ΔE , which is
FWHM
defined as a full width at half maximum (FWHM) of the energy transmit-
tance function, is often used. The second definition considers the relative
energy resolution R¼ΔE/E , or R ¼ΔE /E , where E is the
0 FWHM FWHM 0 0
kinetic energy that corresponds to the average energy to which the energy
analyzeristuned.Sometimestheenergyresolutionisdefinedasavalueequal
to the inverse of R.
Anotherimportantcharacteristicofadeviceintendedforenergyanalysis,
whichinessencedeterminesthesecondconsumerparameterofaspectrom-
eter(namely,thesignal-to-noiseratio),isitscapacityforregisteringthepar-
ticlesemittedbydifferentpartsofanextendedsource.Thiscapacity,called
AdvancesinImagingandElectronPhysics,Volume192 #2015ElsevierInc. 1
ISSN1076-5670 Allrightsreserved.
http://dx.doi.org/10.1016/bs.aiep.2015.08.001
2 Analytical,Approximate-AnalyticalandNumericalMethods
brightness (Kozlov, 1978), can be calculated as the integral of luminosity
taken over the source surface.
Theluminosityofacorpuscular-opticaldeviceisequivalenttothegeomet-
ricalsolidangleΩofparticlecollection.Theso-calledrelativeangleGiscom-
monlyintroduced(Kozlov,1978),whichisdefinedasaratiooftheangleΩto
thefullsolidangle4π:G¼(Ω/4π)100%.Physically,theluminosityexpressesa
ratioofthenumberofparticlesreachingthecollectortothetotalnumberof
particles emitted during the same time by a monoenergetic isotropic point
sourcelocatedattheanalyzerfocus,providedthatthelatterisproperlytuned.
Mathematically,theluminositycanbeexpressedthroughtheplanartilting
anglesα andα ofthegeneratricesoftheconesΩ¼2π(cosα -cosα ),α <α .
1 2 1 2 1 2
2. MAIN TYPES OF ENERGY ANALYZERS
Twotypesofenergyanalyzers—retardinganddispersive—areusedin
the spectroscopy of charged particles.
One basic energy analyzer of the retarding type is a so-called quasi-
sphericalcapacitorwithgrids(Figure1;alsoseeWoodruff&Delchar,1986).
The operation principle of the quasi-spherical capacitor is as follows.
Undertheactionofexcitingradiation(labeled1inthefigure;forexample,
ofelectronsemittedbytheelectrongunEGwithenergyE )thesample
max
(2) under study, which is placed at the geometrical center of the capacitor,
starts to emit the charged particles that travel along the radial trajectories
(3). The grounded grid (4) creates a field-free drift space between the
sample and the area of retarding. The grid (5) is supplied with a retarding
voltage–V .Thecollector(6)iskeptatthepotentialof200–300Vrelative
T
EG
A
6
5 1 +V
const
3
4 –VT
2
Figure 1 Quasi-spherical capacitor scheme: 1—exciting radiation, 2—sample, 3—
secondaryparticles,4—firstgrid,5—retardinggrid,6—collector,EG—electrongun.