Many-body theory of gamma spectra from positron–atom annihilation

22 Pages·2006·0.31 MB·English

INSTITUTEOFPHYSICSPUBLISHING JOURNALOFPHYSICSB:ATOMIC,MOLECULARANDOPTICALPHYSICS
J.Phys.B:At.Mol.Opt.Phys.39(2006)1647–1668 doi:10.1088/0953-4075/39/7/008
Many-body theory of gamma spectra from
positron–atom annihilation
LJMDunlopandGFGribakin
DepartmentofAppliedMathematicsandTheoreticalPhysics,Queen’sUniversity,
BelfastBT71NN,UK
E-mail:[email protected]@qub.ac.uk
Received15December2005,infinalform16December2005
Published13March2006
Onlineatstacks.iop.org/JPhysB/39/1647
Abstract
Amany-bodytheoryapproachtothecalculationofgammaspectraofpositron
annihilation on many-electron atoms is developed. We evaluate the first-
order correlation correction to the annihilation vertex and perform numerical
calculationsforthenoblegasatoms. Extrapolationwithrespecttothemaximal
orbital momentum of the intermediate electron and positron states is used
to achieve convergence. The inclusion of correlation corrections improves
agreementwithexperimentalgammaspectra.
(Somefiguresinthisarticleareincolouronlyintheelectronicversion)
1. Introduction
Themainaimofourpaperistodevelopamany-bodytheoryapproachtothecalculationof
gammaspectrafrompositron–atomannihilation.
Positron annihilation has been an important tool for studying electronic and atomic
structures of solids for over half a century. Its history can be traced from the first review
of the theory of positron annihilation in solids by Ferrel (1956) to a more recent work by
Puska and Nieminen (1994). Positron lifetimes and spectra of annihilation gamma quanta
contain information about the structure and composition of bulk materials and surfaces,
presence and concentration of defects and voids or pores, and their sizes, and electron
momentumdistribution. Interpretationofpositronannihilationdatarequiresgoodtheoretical
understanding of the process. Many-body theory was the first method used to determine
positron annihilation rates in metals (Kahana 1963, Carbotte 1967). In particular, it was
successfulinexplaininglargeenhancementfactorsthatincreasetheannihilationratesabove
thatobtainedinthenoninteractingelectrongasapproximation(seealsoArponenandPajanne
(1979)).
In the 1990s measurements of gamma-ray spectra from positron annihilation on atoms
and molecules in the gas phase became possible (Tang et al 1992, Coleman et al 1994).
0953-4075/06/071647+22$30.00 ©2006IOPPublishingLtd PrintedintheUK 1647
1648 LJMDunlopandGFGribakin
For He the experiment showed excellent agreement with the calculated spectrum obtained
from an elaborate variational positron–He wavefunction (Van Reeth et al 1996). This work
alsorevealedsizeabledeviationsofthespectrumfromaGaussian,whichisoftenusedasan
approximation. For noble gas atoms of Ar, Kr and Xe, a careful study of the shape of the
511 keV gamma-ray line provided an estimate of the contribution of positron annihilation
withinner-shellelectrons(Iwataetal1997b). Alargesystematicstudywasconductedfora
variety of inorganic molecules, alkanes, alkenes, aromatics and perfluorinated and partially
fluorinatedhydrocarbons(Iwataetal1997a). Inparticular,thisworkdeterminedtherelative
probabilityofannihilationonthefluorineatomsandontheC–Hbondsforpartiallyfluorinated
hydrocarbons. Suchinformationisimportantforachievingbetterunderstandingofverylarge
positronannihilationratesonpolyatomicmolecules(see,e.g.,Surkoetal(2005)).
Onthetheoryside,muchprogressinunderstandingtheinteractionoflow-energypositrons
with many-electron atoms has been obtained by using many-body theory. Its application
to positron–atom collisions was pioneered by Amusia et al (1976) who considered positron
scatteringfromHe. Subsequently,theeffectofvirtualpositronium(Ps)formationonpositron–
atom interaction was investigated for heavier noble gas atoms (Dzuba et al 1993, 1996).
TheseworksshowedthatvirtualPsformationgivesalargecontributiontothepositron–atom
attraction1. Itleadstotheemergenceofpositron–atomvirtualandboundstates(Dzubaetal
1995, 1996), and completely alters the picture of low-energy positron–atom scattering
(GribakinandKing1996). Thesepapersemployedanapproximatemethodofaccountingfor
virtualPsformation. Recently,aconsistentmany-bodytheoryapproachhasbeendeveloped.
It is based on the summation of the electron–positron ladder diagram series (Ludlow 2003,
GribakinandLudlow2004). Itscontributionisespeciallyimportantforthecalculationofthe
annihilationrate. Theseandearliercalculations(Dzubaetal1993,1996)havedemonstrated
thatelectron–positroncorrelationeffectscanenhancethepositron–atomannihilationrateas
muchas103times.
InthepaperbyIwataetal(1997b),thefractionofannihilationwithinner-shellelectrons
wasderivedbyfittingtheexperimentaldatawithalinearcombinationofthegammaspectra
forthevalenceandinnershells. Thesespectrawerecalculatedinthesimplestapproximation,
by using the positron wavefunction in the static field of the atom. In spite of a complete
neglect of electron–positron correlations, the shapes of the measured gamma spectra were
described reasonably well. On the other hand, some discrepancies between the theoretical
andexperimentalspectrawereobvious,addinguncertaintytotheestimatesoftheinner-shell
annihilationfractions. Forexample,thedataforArwerecompatiblewithazerocontribution
ofannihilationwith2sand2pelectrons.
Sincecorrelationeffectsplaysuchalargeroleinpositron–atominteractions, theymust
beincludedinthepropertheoryofannihilationgammaspectra. Inwhatfollows,webriefly
recount the main facts and formulae concerning positron annihilation and gamma spectra
using the formalism of creation–destruction operators (section 2). We then proceed to
derive the many-body diagrammatic expansion of the annihilation amplitude, and consider
thesignificanceofvarioustermsandtheirrelationtothetotalannihilationrate(section3). In
section4,expressionsforthezeroth-andfirst-ordercontributionsarereducedtoproductsof
radialmatrixelements, whichcanbeevaluatednumerically. Section5reportstheresultsof
suchcalculationsforthenoblegasatoms,whichconfirmthatthecorrelationcorrectionstothe
annihilationamplitudehaveamarkedeffectonthegammaspectra. Wealsodemonstratethe
1 Thisexplainedthesuccessofearlierpolarized-orbitalcalculationsindescribingpositronelasticscatteringand
annihilationfornoblegasatoms(McEachranetal1980andreferencestherein). Inthatapproximation,thepositron
wastreatedasaheavyparticle,andthestrengthofthepositron–atompolarizationpotentialwasoverestimated,making
upforthecompleteneglectofvirtualPsformation.
Many-bodytheoryofpositronannihilationspectra 1649
importanceofextrapolationoverthemaximalorbitalangularmomentumoftheintermediate
electron and positron states. In the appendix, expressions for the annihilation amplitudes
involvingmany-particlewavefunctionsincoordinatespacearegiven.
2. Basictheory
2.1. Annihilationoperator
Duetotheconservationofmomentum,annihilationofanelectron–positronpairmustleadto
theemissionofatleasttwophotons2. Infact,accordingtoquantumelectrodynamics(QED),
annihilationintotwophotonsispossibleonlyifthetotalspinSofthepairiszero(Berestetskii
etal1982). ForS =1theannihilationresultsintheemissionofthreephotons3.
InQEDtheprocessofelectron–positronannihilationisdescribedbythesecond-orthird-
order diagram, depending on the number of photons emitted (Berestetskii et al 1982). The
photonscarryawaytheenergyoftheparticles,∼2mc2,andhavelargemomenta∼mc,where
m is the electron mass, and c is the speed of light. Intermediate electron/positron states in
theQEDdiagramshavesimilarmomenta. Theyaremuchlargerthanthetypicalmomentaof
valenceandinner-shellatomicelectrons(withtheexceptionof1selectronsinheavyatomswith
Z ∼ 100),orthepositronmomentuminatypicalexperiment4. Asaresult,theannihilation
amplitudeisindependentoftheelectronandpositronmomenta. Hence,theeffectiveoperator
forannihilationintogammaquantawiththetotalmomentumPisproportionalto
(cid:1)
aˆ bˆ δ , (1)
k1 k2 k1+k2,P
(cid:2) (cid:3)k1,k2
whereaˆ bˆ isthedestructionoperatoroftheelectron(positron)withmomentumk (k ),
k1 k2 1 2
andtheδ-functionensuresmomentumconservation,P=k +k (Ferrel1956). Usingelectron
1 2
andpositrondestructionoperatorsψˆ(r)andϕˆ(r)inthecoordinaterepresentation
(cid:4)
1
aˆ = √ e−ik·rψˆ(r)dr, (2)
k
V
(cid:4)
1
bˆ = √ e−ik·rϕˆ(r)dr, (3)
k
V
whereV isthenormalizationvolume,theannihilationoperator(1)canbere-writtenas(Ferrell
1956,Chang1957)
(cid:4)
Oˆ (P)≡ e−iP·rψˆ(r)ϕˆ(r)dr. (4)
a
This equation shows that the annihilation of a nonrelativistic electron–positron pair occurs
whentheparticlesareatthesamepoint. Physically,thiscanbeexplainedusingtheuncertainty
principle. Indeed,thespatialseparationbetweenthepointswherethephotonsareproducedis
abouth¯/mc. ThisismuchsmallerthantheBohrradiusa =h¯2/me2,i.e. thetypicalatomic
0
size,orthedeBrogliewavelengthoftheparticlesinvolved(c = 137inatomicunits,where
h¯ =m=|e|=1).
2 Whenthisprocessoccursinanexternalfield,e.g.,whentheelectronisboundinanatom,annihilationintoasingle
quantumisalsopossible.However,itsprobabilityissmall;see,e.g.,Johnsonetal(1964).
3 ForS = 0annihilationinto2,4,...photonsispossible,whileforS = 1thenumberofphotonsmustbeodd
(3,5,...).Inbothcasestheprocesswiththesmallestnumberofphotonsdominates.
4 Inpositronannihilationwithmatter, eveniftheinitialpositronisfast, itquicklylosesenergyduetoinelastic
ionizingcollisions,andmostoftheannihilationeventsinvolveslowpositrons.
1650 LJMDunlopandGFGribakin
In equations (1) and (4) electron and positron spin indices have been suppressed. The
operator can be used in this form if the total spin of the electron subsystem is zero. In
this case the annihilation rate is equal to that averaged over the positron spin. The rate of
annihilationintophotonswiththetotalmomentumPisgivenbythesquaredmodulusofthe
transitionamplitudeofoperator(4). Thetotalannihilationrateisobtainedbyintegrationover
d3P/(2π)3andsummationoverthefinalstatesofthesystem. Thecorrectabsolutemagnitude
ofthe2-photonand3-photonannihilationratescanbedeterminedbycomparisonwiththespin-
averagedpositroniumannihilationrates(Berestetskiietal1982). Inthefirstcase,thesquared
amplitudemustbemultipliedbyπr2c,whileinthesecondcase—by[4(π2−9)/3]r2αc,where
0 0
r =e2/mc2istheclassicalelectronradius,andα =e2/h¯cisthefinestructureconstant. This
0
shows that 3-photon annihilation is about 103 times slower than 2-photon annihilation, and
canoftenbeneglected.
2.2. Spectrumofphotonsandannihilationrates
Let |i(cid:4) be the state of N electrons and the positron before the annihilation, and |f(cid:4)—the
stateof(N −1)electronsaftertheannihilation. Intwo-photonannihilation,themomentum
distributionofthephotons,
W (P)=πr2c|(cid:5)f|Oˆ (P)|i(cid:4)|2, (5)
f 0 a
determinestheirenergyspectrum. Thetotalenergyofthetwophotonsis
E +E =2mc2+E −E , (6)
γ1 γ2 i f
whereE andE aretheenergiesoftheinitialandfinalstates(notincludingtherestenergyof
i f
theconstituentparticles),andtotalphotonmomentumisp +p =P.
γ1 γ2
For P = 0 the photons are emitted in the opposite direction, p = −p , and have
γ1 γ2
equal energies, E = E = mc2 + 1(E −E ) ≡ E . For P (cid:6)= 0 the photon energy is
γ1 γ2 2 i f γ
Doppler shifted, e.g., for the first photon E = E +Vmccosθ, where V = P/2m is the
γ1 γ
centre-of-massvelocityoftheelectron–positronpair,θ istheanglebetweenthedirectionof
thephotonandV, andweassumethatV (cid:7) c andp = E /c ≈ mc. Hence, theshiftof
γ1 γ1
thephotonenergyfromthecentreoftheline,(cid:8) =E −E ,is
γ1 γ
Pc
(cid:8) = cosθ, (7)
2
andthephotonenergyspectrumisgivenby
(cid:4) (cid:5) (cid:6)
1 d3P
w((cid:8))= W (P)δ (cid:8)− Pccosθ . (8)
f 2 (2π)3
Usingpolarcoordinatesandaveragingoverthedirectionofemissionofthephoton,oneobtains
(cid:4)(cid:4)
1 ∞ P dP d(cid:9)
w((cid:8))= W (P) P. (9)
c 2|(cid:8)|/c f (2π)3
Ontheotherhand,usingCartesiancoordinatesandchoosingthez-axisalongthedirection
ofthephoton,oneobtainsfrom(8)
(cid:4)(cid:4)
2 dP dP
w((cid:8))= W (P ,P ,2(cid:8)/c) x y. (10)
c f x y (2π)3
This form shows that the energy spectrum is proportional to the probability density for a
component of the momentum P. Equation (10) allows one to link w((cid:8)) to the quantity
measured by the angular correlation of annihilation radiation (ACAR) technique. In ACAR
onemeasuresthesmallangle(cid:10)betweenthedirectionofonephotonandtheplanecontaining
Many-bodytheoryofpositronannihilationspectra 1651
theotherphoton5. Ifthedirectionperpendiculartotheplaneisx, then(cid:10) = P /mc. Given
x
thatthedistributionsofP ,P andP areidentical(inanisotropicsystem), oneobtainsthe
x y z
distributionof(cid:10)fromw((cid:8))byasimplechangeofvariable,(cid:10)=2(cid:8)/mc2.
When a low-energy positron annihilates with a bound electron whose energy is ε ,
n
the centre of the photon spectrum is shifted by ε /2 relative to mc2. The width of the
n
two-photon momentum distribution is determined by the typical momenta of the bound
electron, P ∼ (2m|ε |)1/2. The corresponding Doppler width of the annihilation spectrum,
n
(cid:8) ∼ Pc ∼ (|ε |mc2)1/2, is much greater than its shift. Hence, one can regard the line as
n
centredonE =mc2 =511keV,evenfortheannihilationoninner-shellelectrons.
γ
Thetotalannihilationrateinthestate|i(cid:4)isobtainedbyintegrationoverthemomentum
Pandsummationoverthefinalstates
(cid:4)
(cid:1)
d3P
λ= W (P)
f (2π)3
f
(cid:4)
(cid:1)
d3P
=πr2c (cid:5)i|Oˆ†(P)|f(cid:4)(cid:5)f|Oˆ (P)|i(cid:4) . (11)
0 a a (2π)3
f
(cid:7)
Usingclosureinthesubspaceof(N−1)-electronstates, |f(cid:4)(cid:5)f|=1,substituting(4)and
f
integratingoverP,wehave
(cid:4)
λ=πr02c (cid:5)i|nˆ−(r)nˆ+(r)|i(cid:4)dr, (12)
where nˆ−(r) = ψˆ†(r)ψˆ(r) and nˆ+(r) = ϕˆ†(r)ϕˆ(r) are the electron and positron density
operators. Theannihilationrateisthusgivenbytheexpectationvalueoftheelectrondensity
atthepositronintegratedoverthepositroncoordinates.
Equation(12)givesthetwo-photonannihilationrateforaboundpositron–atomstate|i(cid:4).
Theannihilationrateforapositive-energypositronmovingthroughagasofatomsisgivenby
asimilarformula(Fraser1968)
λ=πr2cnZ , (13)
0 eff
wherenisthenumberdensityofthegas,and
(cid:4)
Zeff = (cid:5)i|nˆ−(r)nˆ+(r)|i(cid:4)dr. (14)
Herethestate|i(cid:4)describesacollisionbetweenthepositronwithmomentumkandtheatom
(usuallyinthegroundstate). Itisnormalizedtothepositronplanewaveeik·ratlargepositron–
atomseparations, i.e.,toonepositronperunitvolume. ThedimensionlessparameterZ is
eff
interpretedastheeffectivenumberoftargetelectronswhichcontributetotheannihilation.
If one neg√lects the interaction between the positron and the atom and writes the initial
state as |i(cid:4) = Vb†|0(cid:4), where |0(cid:4) is the ground state of the N-electron atom, equation (14)
k
gives Z = N. In reality, the values are Z can be very different from the actual number
eff eff
ofelectrons(see,e.g.,Surkoetal(2005)). Repulsionfromthenucleuspreventsthepositron
frompenetratingintotheatom, suppressingtheprobabilityofitsannihilationwiththeinner
electrons. Ontheotherhand,thelong-rangepositron–atomattractionincreasesthepositron
density near the atom, making Z larger. The Coulomb attraction within the annihilating
eff
electron–positronpairincreasesZ further(Dzubaetal1996,GribakinandLudlow2004).
eff
5 Thisisso-called1D-ACAR.Amoreadvancedtechnique,2D-ACAR,involvesmeasuringtheanglebetweenthe
dir(cid:8)ectionsofthephotons, i.e. twoprojectionsofthemomentum. Thecorrespondingdistributionisproportional
to Wf(Px,Py,Pz)dPz,andisusefulforstudyingtheelectrondensityandmomentumdistributioninanisotropic
systems,e.g.,solids(PuskaandNieminen1994).
1652 LJMDunlopandGFGribakin
P P
ε ε
n µ
(a) (b)
Figure1. Annihilationamplitudeforapositroninstateεandelectroninstatenoccupiedinthe
atomicgroundstate(a),oranexcitedelectroninstateµ(b). Thedoubledashedlinecorresponds
totheannihilationgammaquantawithmomentumP.
Note that the rate (13) is related to the positron–atom annihilation cross section σ by
a
λ = σ nv, wherev istheincidentpositronvelocity, sothatσ = πr2Z c/v. Ontheother
a a 0 eff
hand,equation(14)showsthatZ hastheformofatransitionamplitudebetweentwoidentical
eff
states|i(cid:4). Hence,aperturbationseriesexpansionforthisquantitycanbedeveloped(Dzubaetal
1993).
3. Many-bodytheory
3.1. Basesandbuildingblocks
Whenstudyingpositron–atominteractionsbymany-bodytheorymethods,itisconvenientto
usethebasesofelectronandpositronstatesoftheHartree–FockHamiltonianoftheground-
state atom. Denoting the corresponding electron (positron) wavefunctions and destruction
operatorsbyψ (ϕ )andaˆ (bˆ ),respectively,wehave
µ ν µ ν
(cid:1) (cid:1)
ψˆ(r)= ψ (r)aˆ , ϕˆ(r)= ϕ (r)bˆ . (15)
µ µ ν ν
µ ν
Substitutinginto(4)weobtaintheeffectiveannihilationoperatorintheform
(cid:1)(cid:1)
Oˆ (P)= (cid:5)P|δ|µν(cid:4)aˆ bˆ , (16)
a µ ν
µ ν
where
(cid:4)
(cid:5)P|δ|µν(cid:4)= e−iP·rψ (r)ϕ (r)dr
µ ν
(cid:4)
≡ e−iP·(r1+r2)/2δ(r1−r2)ψµ(r1)ϕν(r2)dr1dr2. (17)
Thesecondformexplainsthepresenceofδ inournotationfort(cid:7)heamp(cid:7)litude. N(cid:7)otethatthe
sumoverµinequation(16)containstwodistinctcontributions, = + . The
µ µ(cid:1)F µ>F
first sum includes the electron states occupied in the atomic ground state, i.e., those at or
below the Fermi level F (‘holes’). The corresponding terms in the operator (16) describe
positronannihilationwhichleadstocreationofahole, asshownbydiagram(a)infigure1.
ThesecondsumisovertheelectronstatesabovetheFermilevel(‘particles’), anddescribes
positronannihilationwithanexcitedelectron,diagram(b)infigure1.
Substituting(15)andtheiranaloguesforψˆ†(r)andϕˆ†(r)intonˆ−(r)andnˆ+(r),oneobtains
theelectron–positroncontactdensityoperatorfromequation(12)or(14)intheform
(cid:4)
(cid:1)(cid:1)
nˆ−(r)nˆ+(r)dr= (cid:5)ν(cid:10)µ(cid:10)|δ|µν(cid:4)aˆν†(cid:10)aˆµ†(cid:10)aˆµaˆν, (18)
µ,µ(cid:10) ν,ν(cid:10)
Many-bodytheoryofpositronannihilationspectra 1653
P P
P ε ν ε
ε
+ µ +
n n
n
A0 A1 A2
P P P
ε ε ε ν
n
2
+ n + n + µ n + ...
1
n
1
A3 A4 A5
Figure2. Many-bodytheoryexpansionofthepositron–atomannihilationamplitudeleadingto
thefinalstatewithaholen. Thelinestartingwithεcorrespondstothepositron; ν andµare
theintermediatepositronandelectronstates,respectively.Wavylinescorrespondtotheelectron–
positronorelectron–electronCoulombinteractions.
whereµ,µ(cid:10) andν,ν(cid:10) aretheelectronandpositronstates,respectively. Theamplitude
(cid:4)
(cid:5)ν(cid:10)µ(cid:10)|δ|µν(cid:4)= ϕ∗(r )ψ∗(r )δ(r −r )ψ (r )ϕ (r )dr dr (19)
ν(cid:10) 2 µ(cid:10) 1 1 2 µ 1 ν 2 1 2
corresponds diagrammatically to a point-like δ vertex with two positron and two electron
(particleorhole)lines. Itisrelatedtotheannihilationamplitude(17)by
(cid:4)
d3P
(cid:5)ν(cid:10)µ(cid:10)|δ|µν(cid:4)= (cid:5)ν(cid:10)µ(cid:10)|δ|P(cid:4) (cid:5)P|δ|µν(cid:4). (20)
(2π)3
Graphically,thisequationisequivalenttoconnectingthevertices(a)or(b)fromfigure1with
their complex conjugates (mirror images) and ‘pulling them together’ by the double-dashed
lines, to form a δ vertex with four external lines. This basic relation allows one to relate
productsoftermsinthediagrammaticexpansionoftheannihilationamplitude(cid:5)f|Oˆ (P)|i(cid:4)to
a
thecontributionstothetotalannihilationrate,alongthelinesofequations(11)and(12)(see
below).
3.2. Annihilationamplitudes
Consideraprocessoftwo-photonpositronannihilationwithaground-stateatom
e++A−→A++2γ. (21)
Thesimplestfinalstatef ofthepositiveionA+ isthatofanatomwithaholeinanelectron
orbitaln. Inthelowestordertheamplitudeofthisprocessisshownbydiagram(a),figure1.
The exact amplitude (cid:5)f|Oˆ (P)|i(cid:4) can be represented by a many-body theory expansion in
a
powersoftheresidualelectron–positronandelectron–electronCoulombinteractions. Figure2
shows the corresponding zeroth- and first-order diagrams, together with the main types of
second-orderdiagrams.
Analytically, the first diagram in figure 2 is equal to (cid:5)P|δ|nε(cid:4) (cf equation (17)).
Expressionsforthehigherorderdiagramsareconstructedusingstandardatomicmany-body
theory diagrammatic rules (see, e.g., Amusia and Cherepkov (1975)), modified to account
fortheattractiveelectron–positronCoulombinteraction. Theseexpressionscontainproducts
1654 LJMDunlopandGFGribakin
of Coulomb and δ-function matrix elements divided by energy denominators. Each energy
denominatorcorres(cid:7)pondsto(cid:7)anintermediatestateseparatingtwointeractionsinthediagram.
ItisgivenbyE− ε + ε +i0,wherethesumsareoverallparticles6 andallholes
part hole
intheintermediatestate,Eistheenergyenteringthediagramandtheinfinitesimalimaginary
i0 defines the way of bypassing the pole. Summation over all intermediate positron and/or
electronstatesiscarriedout. Theoverallsignfactorforadiagramisgivenby(−1)Nh+Nl+Nep,
whereN ,N andN arethenumbersofinternalholelines,electron–holeloopsandelectron–
h l ep
positroninteractions,respectively.
Forexample,thefirst-orderdiagramA1infigure2correspondsto
(cid:1)(cid:5)P|δ|µν(cid:4)(cid:5)νµ|V|nε(cid:4)
− , (22)
ε−ε −ε +ε
µ,ν ν µ n
where
(cid:4)
1
(cid:5)νµ|V|nε(cid:4)= ϕ∗(r )ψ∗(r ) ψ (r )ϕ (r )dr dr (23)
ν 2 µ 1 |r −r | n 1 ε 2 1 2
1 2
istheCoulombmatrixelement. DiagramA1representsacorrectiontotheannihilationvertex
(diagramA0)duetotheelectron–positroninteraction. Notethatforlowpositronenergiesthe
denominatorinequation(22)isneverzero(i.e.,theintermediatestateisvirtualforanyε and
ν
ε );hencei0hasbeendropped.
µ
The second-order diagrams shown in figure 2 have a clear physical interpretation.
Thus, diagram A2 represents a higher order correction to the annihilation vertex, similar
todiagramA1,withanextrainteractionbetweentheannihilatingelectronandpositron. Such
contributions are very important because the electron–positron Coulomb attraction strongly
enhancestheamplitudeoffindingthetwoparticlesatthesamepoint. DiagramA3describes
polarizationoftheatombythepositron. Thispolarizationgivesrisetoanattractivepositron–
atomcorrelationpotentialwhichbehavesas−α e2/2r4atlargedistances,α beingtheatomic
d d
dipole polarizability. It has a large effect on the incident positron (see, e.g., Dzuba et al
(1993, 1996)). DiagramA4isacorrelationcorrectiontotheHartree–Fockwavefunctionof
thehole. GiventhattheHartree–Fockapproximationdescribesatomicground-stateorbitals
well,onemayexpectsuchcorrectionstoberelativelysmall. Finally,diagramA5belongsto
anothertypeofvertexcorrectionsinwhichannihilationtakesplaceinthepresenceofavirtual
hole–particleexcitation. Wewillshowbelowthatthisandsimilardiagramscanbeneglected
becauseofaspecificcancellationbetweenthecontributionsofcertainpairsofdiagramsinthe
photonmomentumdistribution.
As explained at the end of section 2.2, Z can also be presented as a many-body
eff
perturbation series. If the state |i(cid:4) in equation (14) is that of a positron incident on the
ground-stateatom,Z isgivenbythediagrammaticexpansionshowninfigure3. Eachofthe
eff
diagramscontainsaδ-functionvertexcorrespondingtothecontactdensityoperator(18)with
matrixelements(19). Thezeroth-orderdiagramZ0describesannihilationofthepositronwith
anelectroninaHartree–Fockorbitaloccupiedinthegroundstate. Itsanalyticalexpression,
(cid:4)
(cid:1) (cid:1)
(cid:5)εn|δ|nε(cid:4)= |ψ (r)|2|ϕ (r)|2dr, (24)
n ε
n n
6 Iftheannihilationphot(cid:7)onsrepresentedbythedouble-dashedlinearepresentintheintermediatestate,theirenergy
mustalsobeincludedin εpart. Sincetheelectronandpositronenergiesdonotincludetheirrestenergymc2,the
energyofthephotonsmustbegivenwithrespectto2mc2.
Many-bodytheoryofpositronannihilationspectra 1655
ε ε ε ν ε ε ε ε ε ε ε
µ
+ + + +
n n
n n n
Z0 Z1 Z2 Z3 Z4
ε ε ε ε ε ε ε ε
+ + + + + ...
Z5 Z6 Z7 Z8
Figure3. Many-bodytheoryexpansionofthepositron–atomannihilationrateZeff. Diagrams
Z1–Z8arecorrectionstothezeroth-orderannihilationvertex, diagramZ0. Correctionstothe
externalpositronlines(ε)arenotshown.
contains contributions of all occupied electron states n. Higher order diagrams in figure 3
represent correlation corrections to the annihilation vertex. Their analytical expressions are
derivedusingtherulesformulatedabove. Forexample,diagramZ1correspondsto
(cid:1) (cid:5)εn|δ|µν(cid:4)(cid:5)νµ|V|nε(cid:4)
− , (25)
ε−ε −ε +ε
µ,ν,n ν µ n
cfequation(22). Notethatinfigure3wedonotshowcorrectionstotheexternalpositronlines
ε,likethatindiagramA3offigure2. Thesecorrectionscanbeeasilyincludedbyreplacing
thepositronstateϕ inthestaticfieldoftheatombythepositronDysonorbitalwhichaccounts
ε
for the positron–atom correlation potential (Dzuba et al 1996, Gribakin and Ludlow 2004).
Byanalogywithequations(11)–(12)and(14),thetotalannihilationrateisrelatedtothe
momentumdistributionofthephotonpairsby
(cid:4)
(cid:1)
d3P
Z = |(cid:5)f|Oˆ (P)|i(cid:4)|2 . (26)
eff a (2π)3
f
Thismeansthatifthetermsoftheseriesinfigure2aremultipliedbytheircomplexconjugates
and integrated over P, the expansion for Z from figure 3 must be recovered. Indeed, it is
eff
easy to see, using equation (20), that the contribution of diagram A0 to the right-hand side
of equation (26) (i.e., A0 times its complex conjugate A0∗) is equal to diagram Z0, after
summationovern. Similarly,theproductofdiagramsA1andA0∗,
(cid:9) (cid:10)
(cid:4)
d3P (cid:1)(cid:5)P|δ|µν(cid:4)(cid:5)νµ|V|nε(cid:4) (cid:1)(cid:5)εn|δ|µν(cid:4)(cid:5)νµ|V|nε(cid:4)
(cid:5)εn|δ|P(cid:4) − =− , (27)
(2π)3 ε−ε −ε +ε ε−ε −ε +ε
µ,ν ν µ n µ,ν ν µ n
givesZ1(aftersummationovern),whiletheproductofA1andA1∗becomesZ3.
However, somecontributionsthatappearontheright-handsideofequation(26)donot
match any diagrams of the Z expansion. In fact, their analytical expressions do not even
eff
have the form that would identify them with a particular diagram. For example, consider a
productofdiagramsA5andA0∗fromfigure2
(cid:1) (cid:5)P|δ|n ν(cid:4)(cid:5)n n |V|nµ (cid:4)(cid:5)νµ |V|n ε(cid:4)
(cid:5)εn|δ|P(cid:4) (cid:2) 2 1 2 (cid:3)(cid:2)1 1 1 (cid:3), (28)
ε−ε −ε +ε +ε ε−ε −ε +ε
νµ1n1n2 2γ µ1 n1 n2 ν µ1 n1
where ε = ε+ε is the energy of the two gamma quanta with respect to 2mc2, given by
2γ n
theenergyconservation,E +E = 2mc2+ε+ε . Itiseasytocheckthattheexpression
γ1 γ2 n
1656 LJMDunlopandGFGribakin
P P P
ε ν ε ε
µ
n n
µ21 + µn2 + µn21 + ...
1
n1 n n1
1
B1 B2 B3
Figure4. Amplitudeofpositronannihilationleadingtothefinalstatewithtwoholes,n1andn2,
andexcitedelectronµ1.Forn1(cid:6)=n2,exchangediagramswiththeindicesn1andn2swappedand
anextrafactorof−1mustalsobeconsidered.
obtained upon integration of (28) over P cannot be drawn as a diagram consistent with the
diagrammatic rules. Hence, it also does not correspond to any diagram in the expansion of
Z .
eff
Tosolvethisparadox,notethatequation(26)containsasumoverallfinalstates. Thus,
besidestheprocessshowninfigure2,wherepositronannihilationleadstoasingle-holestate,
one must consider processes with other types of final states. Their formation is a result of
extraelectron–positronorelectron–electroncorrelations(whichmeansthattheircontribution
tothetotalannihilationspectrumshould,ingeneral,besmaller). Thesimplestofsuchstates
containstwoholesandoneexcitedelectron. Thecorrespondingamplitudeinthelowest(first)
orderisshowninfigure4.
It is easy to see how these diagrams contribute to the right-hand side of equation (26).
Thus,B1timesB1∗(integratedoverP)becomesdiagramZ7,B1timestheexchangeanalogue
of B1∗ results in diagram Z8, while B2 times B2∗ gives diagram Z5. At the same time, B1
timesB3∗,
(cid:1)(cid:5)n n |V|nµ (cid:4)(cid:5)εn|δ|P(cid:4)(cid:5)P|δ|n ν(cid:4)(cid:5)νµ |V|n ε(cid:4)
1 2(cid:2) 1 (cid:3)(cid:2) 2 1 (cid:3) 1 , (29)
ε−ε +ε ε−ε −ε +ε
νn 2γ n ν µ1 n1
does not lead to any Z diagram after integration over P. In fact, it does not lead to any
eff
valid diagrammatic expression at all. However, the matrix elements in equation (29)
match those in equation (28). Using the energy conservation for the process in figure 4,
ε = ε + ε − ε − ε , we see that the first energy denominator in (29) is equal to
2γ µ1 n1 n2
ε +ε −ε −ε ,whilethefirstenergydenominatorin(28)isequalto−ε −ε +ε +ε .
n µ1 n1 n2 n µ1 n1 n2
Hence,thetwoexpressionscanceltermbyterm,as(28)issummedovertheholestatesnand
(29)issummedovertheelectronandholestatesν ,n andn .
1 1 2
Note that this cancellation of the two anomalous contributions to the total annihilation
rate, Z , is exact. However, it occurs even before the integration over the momentum of
eff
thephotons. Thismeansthatthesecontributionscanbeomittedwhencalculatingthephoton
momentum distribution and gamma spectrum. In particular, diagram A5 in figure 2 can be
ignored. Asimilar cancellation alsotakes place for other diagrams representing corrections
totheannihilationvertex,inwhichtheincidentpositronlineisconnectedwiththefinal-state
hole by means of Coulomb interactions and electron–hole pairs. In all of these diagrams,
annihilationoccursinthepresenceofoneormorevirtualelectron–holepairs.
In principle, the photon spectra corresponding to different final ionic states (e.g., those
showninfigures2and4)couldbeconsideredseparately. Inthiscase,thecontributions(28)
and (29) are not added together, and do not cancel each other. In practice, though, the total
energy of the two photons or the final state of the ion is usually not observed. The shift of