Elementary Scattering Theory: For X-ray and Neutron Users

214 Pages·2011·2.797 MB·English

by  D.S. Sivia

ELEMENTARYSCATTERINGTHEORY
This page intentionally left blank
Elementary Scattering Theory
For X-ray and neutron users
D.S. SIVIA
StJohn’sCollege,Oxford
1
3
GreatClarendonStreet,OxfordOX26DP
OxfordUniversityPressisadepartmentoftheUniversityofOxford.
ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship,
andeducationbypublishingworldwidein
Oxford NewYork
Auckland CapeTown DaresSalaam HongKong Karachi
KualaLumpur Madrid Melbourne MexicoCity Nairobi
NewDelhi Shanghai Taipei Toronto
Withofficesin
Argentina Austria Brazil Chile CzechRepublic France Greece
Guatemala Hungary Italy Japan Poland Portugal Singapore
SouthKorea Switzerland Thailand Turkey Ukraine Vietnam
OxfordisaregisteredtrademarkofOxfordUniversityPress
intheUKandincertainothercountries
PublishedintheUnitedStates
byOxfordUniversityPressInc.,NewYork
(cid:1)c D.S.Sivia2011
Themoralrightsoftheauthorhavebeenasserted
DatabaserightOxfordUniversityPress(maker)
Firstpublished2011
Allrightsreserved.Nopartofthispublicationmaybereproduced,
storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,
withoutthepriorpermissioninwritingofOxfordUniversityPress,
orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate
reprographicsrightsorganization.Enquiriesconcerningreproduction
outsidethescopeoftheaboveshouldbesenttotheRightsDepartment,
OxfordUniversityPress,attheaddressabove
Youmustnotcirculatethisbookinanyotherbindingorcover
andyoumustimposethesameconditiononanyacquirer
BritishLibraryCataloguinginPublicationData
Dataavailable
LibraryofCongressCataloginginPublicationData
Dataavailable
TypesetbytheauthorinLaTeX
PrintedinSpain
onacid-freepaperby
GraficasEstella
ISBN 978–0–19–922867–6(Hbk.)
978–0–19–922868–3(Pbk.)
1 3 5 7 9 10 8 6 4 2
ToRichardandRoger
This page intentionally left blank
Preface
Theopportunitiesfordoingexperimentsatsynchrotronandneutron
facilitieshavegrownrapidlyinrecentyears,andaresettocontinue
intotheforeseeablefuture. PowerfulthirdgenerationX-raysources,
comprisingthe APS in Chicago,the ESRF in Grenobleand SPring-8
in Hyo¯go, which became operational in the 1990s, are being joined
by an increasing number of new regional facilities; DIAMOND and
SOLEIL,forexample,to namebuttwo. Similarly,theavailability of
neutronsisbeingenhancedbytheupgradeandexpansionofestab-
lishedfacilities,liketheILLinFranceandISISinEngland,andthe
commissioning of next generation sources, such as the SNS in the
UnitedStatesandJ-PARCinJapan.
The early years of X-ray and neutron scattering were dominated
byphysicists,whotraditionallyhadastrongbackgroundintherel-
evant theory and mathematics needed to understand the nature of
theexperiments;eruditetexts,suchasLovesey(1986),couldbecon-
sulted when details were required. The recent rise in the acces-
sibility of synchrotrons and neutrons has been accompanied, if not
driven,byashiftinemphasisfromthestudyofhardcondensedmat-
ter to the investigation of biological samples and soft matter. As a
result,thedemographicsoftheusersattherespectivefacilitieshas
been changing rapidly: an increasing proportion are biologists and
chemists, forwhomthe scholarlyphysics-orientedworksaregener-
allytoodemandingtobehelpful.Thisbookaimstoprovideagentler
introductionto thesubject,tocaterforthosewhoarelessconfident
intheirmathematicalabilityandphysicsbackground.
It could be argued that a knowledge of scattering theory is not
necessaryforbeingauseratasynchrotronorneutronfacilitythese
days, because the techniques are sufficiently well-developed to be
treatedasstandardtoolswheretherequireddataanalysisiscarried
out with tried-and-tested computer programs. While there is truth
inthisviewpoint,abasicunderstandingoftheunderlyingtheoryis
alwaysintellectuallyfulfillinganduseful: iteducatesusaboutwhat
can be inferred reliably from the measurements and, hence, influ-
enceswhichexperimentsaredone. Itisalsoimportanttoremember
thatnoanalysisprogramiseverfool-prooforfreefromassumptions
andapproximations; consequently,it is advantageousto have some
ideaof howthedata are relatedto the objectofinterestso thatthe
numericalresultscanbecheckedagainstcommonsense.
Although mathematics cannot be avoided in a theoreticaldiscus-
sion,my aim hasbeento writea bookthatmost scientists will find
viii Preface
approachable. Tothisend,thefirsttwochaptersaredevotedtopro-
vidingatutorialonbackgroundmaterialthatisimplicitlyassumed
in other texts. Thereafter, the philosophy has been one of keeping
things as simple as possible: clarity and understanding have been
favoured over a comprehensivetreatment, through the omission of
topics and detail that entail too much complexity. Despite my best
efforts, some students will still find this book quite daunting. If
theycangetsomethingoutofit,thenI willconsiderthe endeavour
tohavebeenworthwhile.
Finally,itisagreatpleasuretoacknowledgethehelpandencour-
agement that I received from a number of friends and colleagues.
First and foremost, Doryen Bubeck, Steve Collins, Bill Hamilton,
Jerry Mayers, Jeff Penfold, Dave Waymont and John Webster, who
gavemeusefulfeedbackonmanyofthechapters. Atvariouspoints,
I’ve also picked the brains of Nikitas Gidopoulos, Stephen Lovesey,
Roger Pynn and Jorge Quintanilla. Assistance and advice on indi-
vidualtopicsandchapterswasgivenbyPeterBattle,IainCampbell,
Marilyn Hawley, Steve King, Kevin Knight, Stewart Parker, Owen
SaxtonandMonaYethiraj. ManythanksgotomyEditor,So¨nkeAd-
lung,forhispatience;andtoEmmaLonie,AprilWarmanandother
staffattheOxfordUniversityPress. Last butnotleast,Iamgrate-
fultoUnileverR&Dfortheir generoussponsorship ofthis book,and
toIanTuckerforhishelpinsecuringit.
Oxford
July,2010 D.S.S.
Contents
I Some preliminaries 1
1 Studyingmatterattheatomicandmolecularlevel 3
1.1 Lengthscalesandlogarithmicaxes 3
1.2 Resolution,magnificationandmicroscopy 6
1.3 Structure,dynamicsandspectroscopy 9
1.4 Atomicbuildingblocksandinteractions 11
1.4.1 Thebasicstructureoftheatom 12
1.4.2 Thefundamentalforcesofnature 13
1.4.3 Probingmatterbyscatteringparticles 14
1.5 Energy,lengthandtemperaturescales 16
1.6 Atableofusefulconstants 18
2 Waves,complexnumbersandFouriertransforms 19
2.1 Sinusoidalwaves 19
2.1.1 Thedirectionofpropagation 21
2.1.2 Theprincipleofsuperposition 23
2.2 Complexnumbers 25
2.2.1 Definition 25
2.2.2 Basicalgebra 26
2.2.3 TheArganddiagram 27
2.2.4 Theimaginaryexponential 27
2.3 Fourierseries 29
2.3.1 OrthogonalityandtheFouriercoefficients 30
2.3.2 ThecomplexFourierseries 32
2.4 Fouriertransforms 32
2.4.1 Convolutiontheorem 34
2.4.2 Auto-correlationfunction 36
2.5 Fourieropticsandphysicalinsight 38
2.5.1 Young’sdoubleslit 40
2.5.2 Asinglewideslit 41
2.5.3 Adiffractiongrating 42
2.5.4 Theconvolutiontheoreminaction 43
2.5.5 Multi-dimensionalgeneralization 44
2.6 Fourierdataanalysis 48
2.6.1 Thephaseproblem 48
2.6.2 Truncationeffectsandwindowing 50
2.6.3 Noiseandprobabilitytheory 52