Positron-molecule interactions: Resonant attachment, annihilation, and bound states

51 Pages·2010·2.13 MB·English

REVIEWSOFMODERNPHYSICS,VOLUME82,JULY–SEPTEMBER2010
Positron-molecule interactions: Resonant attachment, annihilation,
and bound states
G. F. Gribakin*
DepartmentofAppliedMathematicsandTheoreticalPhysics,Queen’sUniversityBelfast,
Belfast,BT71NN,NorthernIreland,UnitedKingdom
J. A. Young†
JetPropulsionLaboratory,CaliforniaInstituteofTechnology,4800OakGroveDrive,
Pasadena,California91109,USA
C. M. Surko‡
DepartmentofPhysics,UniversityofCalifornia,SanDiego,9500GilmanDrive,LaJolla,
California92093-0319,USA
(cid:1)Published14September2010(cid:2)
Thisarticlepresentsanoverviewofcurrentunderstandingoftheinteractionoflow-energypositrons
withmoleculeswithemphasisonresonances,positronattachment,andannihilation.Measurementsof
annihilation rates resolved as a function of positron energy reveal the presence of vibrational
Feshbach resonances (cid:1)VFRs(cid:2) for many polyatomic molecules. These resonances lead to strong
enhancement of the annihilation rates. They also provide evidence that positrons bind to many
molecular species. A quantitative theory of VFR-mediated attachment to small molecules is
presented.Itistestedsuccessfullyforselectedmolecules(cid:1)e.g.,methylhalidesandmethanol(cid:2)whereall
modes couple to the positron continuum. Combination and overtone resonances are observed and
theirroleiselucidated.Moleculesthatdonotbindpositronsandhencedonotexhibitsuchresonances
are discussed. In larger molecules, annihilation rates from VFR far exceed those explicable on the
basisofsingle-moderesonances.Theseenhancementsincreaserapidlywiththenumberofvibrational
degreesoffreedom,approximatelyasthefourthpowerofthenumberofatomsinthemolecule.While
thedetailsareasyetunclear,intramolecularvibrationalenergyredistribution(cid:1)IVR(cid:2)tostatesthatdo
not couple directly to the positron continuum appears to be responsible for these enhanced
annihilation rates. In connection with IVR, experimental evidence indicates that inelastic positron
escape channels are relatively rare. Downshifts of the VFR from the vibrational mode energies,
obtained by measuring annihilate rates as a function of incident positron energy, have provided
bindingenergiesfor30species.Theirdependenceuponmolecularparametersandtheirrelationship
to positron-atom and positron-molecule binding-energy calculations are discussed. Feshbach
resonances and positron binding to molecules are compared with the analogous electron-molecule
(cid:1)negative-ion(cid:2) cases. The relationship of VFR-mediated annihilation to other phenomena such as
Dopplerbroadeningofthegamma-rayannihilationspectra,annihilationofthermalizedpositronsin
gases, and annihilation-induced fragmentation of molecules is discussed. Possible areas for future
theoreticalandexperimentalinvestigationarealsodiscussed.
DOI:10.1103/RevModPhys.82.2557 PACSnumber(cid:1)s(cid:2): 34.80.Lx,34.50.(cid:1)s,71.60.(cid:2)z,78.70.Bj
CONTENTS G. Resonantannihilationinlargemolecules 2567
1. Vibrationalleveldensities 2567
2. Mode-basedresonantdoorwaystates 2569
I. IntroductionandOverview 2558
3. AnnihilationandtheonsetofIVR 2570
II. Theory 2561
H. Calculationsofannihilationandbinding 2571
A. Annihilationbasics 2561
1. Annihilation 2571
B. Gamma-rayspectraandannihilationrates 2562
2. Positron-moleculebinding 2573
C. Positron-moleculewavefunction 2563
III. ExperimentalToolsandProcedures 2576
D. Directannihilation:Virtualandweaklybound
A. Annihilation-ratemeasurementswiththermalized
positronstates 2564
positronsinatmosphericpressuregases 2576
E. Resonantannihilation 2566
B. Buffer-gaspositrontrapsastailoredsourcesof
F. Resonancesduetoinfrared-activemodes 2566
positrons 2576
C. Annihilation-ratemeasurementsinpositrontraps 2577
D. Trap-basedcoldpositronbeams 2577
*[email protected] E. Energy-resolvedannihilationmeasurements 2578
†[email protected] F. Gamma-rayspectralmeasurements 2579
‡[email protected] G. Annihilation-inducedfragmentation 2579
0034-6861/2010/82(cid:1)3(cid:2)/2557(cid:1)51(cid:2) 2557 ©2010TheAmericanPhysicalSociety
2558 Gribakin,Young,andSurko: Positron-moleculeinteractions:Resonant…
IV. AnnihilationonSmallMolecules 2579 al.,1992;Churazovetal.,2005;Guessoumetal.,2010(cid:2).A
A. HalomethanesasabenchmarkexampleofVFR 2580 current research goal is the creation of a Bose conden-
B. Methanol:AcaseofmultimodeVFR 2581 sate of positronium (cid:1)Ps(cid:2) atoms (cid:1)i.e., the electron-
C. VFRfromdipole-forbiddenvibrations 2582 positronanalogofthehydrogenatom(cid:2)thatoffersprom-
D. EffectofmolecularsizeonthemagnitudesofVFR 2583 ise for the development of an annihilation gamma-ray
E. Nonresonantannihilationinsmallmolecules 2583 laser (cid:1)Mills, 2002, 2007; Mills et al., 2004; Cassidy and
F. Smallmoleculesummary 2585 Mills, 2007(cid:2).
V. IVR-EnhancedResonantAnnihilationinLarger Typically, positrons from conventional sources (cid:1)e.g.,
Molecules 2585 radioisotopes or electron accelerators(cid:2) slow down from
A. Overview 2585 energies of kilovolts to hundreds of kilovolts to (cid:4)50 eV
B. Thealkanemoleculeparadigm 2586 beforeannihilating.Inthecaseofatomsormolecules,if
C. DependenceofZ onmolecularsize 2587 the incident positron energy (cid:5) is greater than the Ps-
eff
D. Towardamodelofannihilationinlargemolecules 2589 formationthresholdE =E −E ,whereE istheioniza-
th i Ps i
E. Inelasticautodetachment 2589 tion energy of the target and E =6.8 eV is the binding
Ps
1. Fluorine-substitutedalkanes 2589 energy of the ground-state Ps atom, then the dominant
2. EffectsofmoleculartemperatureonZ 2590 annihilationprocessisthroughPsformation.Theresult-
eff
F. OtherIVR-relatedphenomena 2591 ing Ps atom subsequently annihilates by emitting two or
G. Largemoleculesummary 2592 three gamma-ray quanta.
VI. Positron-MoleculeBindingEnergies 2592 In this review, attention is restricted to positron ener-
A. Relationtomolecularproperties 2592 giesbelowthePs-formationthreshold,0(cid:6)(cid:5)(cid:6)E ,where
th
B. Comparisonwithnegativeionsofmoleculesand the Ps channel is closed. Here annihilation occurs as a
clusters 2595 result of the overlap of the positron and electron densi-
VII. AnalysisofAnnihilationRatesMeasuredwith tiesduringthecollision.Thebasicrateinthiscaseisthe
ThermalizedPositrons 2596 Dirac rate (cid:7) for two-gamma annihilation in a free-
D
VIII. OtherTopics 2597 electron gas (cid:1)Dirac, 1930(cid:2)
A. Gamma-rayDoppler-broadeningmeasurements 2597
B. Annihilation-inducedfragmentationofmolecules 2600 (cid:7) =(cid:8)r2cn , (cid:1)1(cid:2)
D 0 e
C. Nonlineardependenceofannihilationonmolecular
wherer istheclassicalelectronradius,r =e2/mc2incgs
density 2601 0 0
IX. SummaryandaLooktotheFuture 2602 units,e andm aretheelectronchargeandmass,c isthe
Acknowledgments 2603 speed of light, and ne is the electron density.
References 2604 In his seminal discovery of the positronium atom,
Deutsch(cid:1)1951a,1951b(cid:2)foundacuriouseffect.Although
the annihilation rate for thermal positrons at 300 K in
I.INTRODUCTIONANDOVERVIEW
atomicandmoleculargaseswasapproximatelyinaccord
with Eq. (cid:1)1(cid:2) for some species (cid:1)e.g., argon and nitrogen(cid:2),
The subject of this review is the interaction of low-
the rate for dichlorofluoromethane CCl F (cid:1)“freon-12”(cid:2)
energy positrons with molecules. Positrons, the antipar- 2 2
wasmuchlarger.Deutschinsightfullyascribedthiseffect
ticles of electrons, are important in many areas of sci-
to some type of resonant positron-molecule attachment
ence and technology. Much of their utility relies on the
process. A decade later, Paul and Saint-Pierre (cid:1)1963(cid:2)
fact that, when an electron and positron interact, they
measuredannihilationratesingasesofalkanemolecules
can annihilate, producing a characteristic burst of
C H , from methane to butane, n=1–4. They found
gamma rays. The lowest order process results in two n 2n+2
back-to-back photons, each with the energy of the rest that the rate (cid:7) was much greater than (cid:7)D and that the
mass of the electron (cid:1)or positron(cid:2) (cid:1)511 keV(cid:2). ratio (cid:7)/(cid:7)D increased exponentially with molecular size.
The annihilation of low-energy (cid:1)e.g., (cid:3)50 eV(cid:2) posi- Annihilationratesingasesareconventionallynormal-
ized to the Dirac rate. The corresponding dimensionless
tronsonatomsandmoleculesplaysaparticularlyimpor-
quantity is the “effective number of electrons”1
tant role in many fields. In medicine, positron emission
tomography (cid:1)PET(cid:2) exploits two-gamma annihilation to (cid:7)
studyhumanmetabolicprocesses(cid:1)Wahl,2002(cid:2).Inmate- Z = , (cid:1)2(cid:2)
eff (cid:8)r2cn
rial science, there are numerous positron-based tech- 0
niques to study the properties of matter (cid:1)Schultz and where n is the density of atoms or molecules (cid:1)Pomeran-
Lynn, 1988; Puska and Nieminen, 1994; Dupasquier and chuk, 1949; Fraser, 1968(cid:2). For a simple collision and ne-
Mills, 1995; Coleman, 2000(cid:2), including the Fermi sur-
glectingelectron-positroncorrelations,onemightexpect
faces in metals (cid:1)Major et al., 2004(cid:2), microscopic pores in
that (cid:7)(cid:3)(cid:7) so that Z is comparable to Z=n /n, the
solids(cid:1)Gidleyetal.,2000,2006(cid:2),thefreevolumeinpoly- D eff e
mers (cid:1)Dlubek et al., 1998(cid:2), and the composition and
structure of surfaces (cid:1)David et al., 2001(cid:2). In astronomy, 1In chemical kinetics, Z corresponds to the (cid:1)normalized(cid:2)
511 keV annihilation radiation (cid:1)the strongest gamma- rate constant of the anniheiflfation reaction. In positron physics
ray line of extraterrestrial origin(cid:2) has been proven to be this quantity is commonly referred to as the “annihilation
useful in elucidating astrophysical processes (cid:1)Ramaty et rate.”
Rev.Mod.Phys.,Vol.82,No.3,July–September2010
Gribakin,Young,andSurko: Positron-moleculeinteractions:Resonant… 2559
total number of electrons per target atom or molecule. Gamma-ray spectra were measured for many molecules
However, values of Z are often much larger (cid:1)e.g., for (cid:1)Iwata, Greaves, and Surko, 1997(cid:2). The Z /Z ratios
eff eff
butane, Z /Z=600(cid:2). werefoundtoincreaserapidlywithmolecularsizeupto
eff
Positronannihilationinatomsandmoleculeswassub- species as large as naphthalene and hexadecane
sequently studied for a wide range of species (cid:1)Osmon, (cid:1)C H (cid:2), reaching values (cid:9)104 (cid:1)Surko, Passner, et al.,
16 34
1965a, 1965b; Tao, 1965, 1970; McNutt et al., 1975; 1988; Murphy and Surko, 1991(cid:2).3
Sharma and McNutt, 1978; Charlton et al., 1980, 2002, A key to further progress was the development of a
2006; Heyland et al., 1985, 1986; Sharma et al., 1985; Al- trap-based positron beam with a narrow energy spread
Qaradawi et al., 2000(cid:2). Early experiments were done (cid:1)(cid:3)40 meV(cid:2)(cid:1)Gilbertetal.,1997;Kurzetal.,1998(cid:2).Using
withthermalpositronsingasesatatmosphericdensities, this beam, annihilation rates for atoms and molecules
n(cid:3)1 amagat (cid:1)Deutsch, 1953; Paul and Saint-Pierre, weremeasuredasafunctionofincidentpositronenergy
1963; Griffith and Heyland, 1978(cid:2).2 Later, experiments
from 50 meV to many electron volts. The result was the
were done at much lower densities using positrons
discovery of resonances associated with the molecular
trappedandcooledto300 K(cid:1)Surko,Passner,etal.,1988;
vibrational modes, namely, vibrational Feshbach reso-
Murphy and Surko, 1991; Iwata et al., 1995; Iwata, nances (cid:1)VFRs(cid:2) (cid:1)Gilbert et al., 2002(cid:2).
Greaves, and Surko, 1997(cid:2). The experiments showed
A crucial point is that VFRs generally require the ex-
that the annihilation rates for many molecular species
istence of a bound state of the positron and the mol-
exceededgreatlythenaivebenchmarkrate,Z (cid:3)Z,and
eff ecule. They occur when the incident positron excites a
a number of chemical trends were identified.
vibrational mode and simultaneously makes a transition
Since Deutsch’s first results, these large annihilation
from the continuum into the bound state. The existence
rates were associated with some kind of resonance phe-
of both low-lying vibrational excitations and a positron
nomenon or attachment process. Goldanskii and Saya-
bound state thus enables the formation of long-lived
sov (cid:1)1964(cid:2) discussed the possibility of resonance-
positron-molecule resonant complexes in a two-body
enhancedannihilationduetoaboundorvirtualpositron
collision. The lifetime of these quasibound states is lim-
stateclosetozeroenergy.SmithandPaul(cid:1)1970(cid:2)consid-
ited by positron autodetachment accompanied by vibra-
ered the possibility that the large annihilation rates in
tional deexcitation. The upper limit on the lifetime is
molecules were due to a vibrational resonance, and sev-
(cid:4)0.1 ns,setbythepositronannihilationrateinthepres-
eral other explanations were proposed (cid:1)Surko, Passner,
ence of atomic-density electrons.
et al., 1988; Dzuba et al., 1996; Laricchia and Wilkin,
1997;Gribakin,2000(cid:2).However,progresswashampered The annihilation rate as a function of positron energy
Z (cid:1)(cid:5)(cid:2) (cid:1)i.e., the “annihilation spectrum”(cid:2) for the four-
greatly by the lack of data other than for positrons with eff
carbon alkane, butane, is shown in Fig. 1 (cid:1)Gilbert et al.,
thermal energy distributions at 300 K. The summary
2002(cid:2). While there is some qualitative correspondence
statement in 1982 by Sir Harrie Massey was that annihi-
between the Z (cid:1)(cid:5)(cid:2) and the infrared (cid:1)IR(cid:2) absorption
lationstudieswere“completelymysteriousatpresentin eff
almostallsubstances”(cid:1)Fraseretal.,1982;Massey,1982(cid:2), spectrumofthemolecule,theshapesofthespectralfea-
tures are quite different (cid:1)Barnes et al., 2003(cid:2).
and this remained more or less correct for another
These positron VFRs can be compared to resonances
20 years.
that play an important role in electron attachment to
In the broader view, processes that are commonplace
molecules and clusters (cid:1)Christophorou et al., 1984; Ho-
inphysicsinvolvingmatter,suchaslow-energytwo-body
scatteringevents,havefrequentlybeenfoundtobefrus- topetal.,2003(cid:2).Theelectroncollisionresultsinthepro-
tratingly difficult to study when antiparticles are in- duction of long-lived (cid:1)metastable(cid:2) parent anions or mo-
volved (cid:1)Schultz and Lynn, 1988; Eades and Hartmann, lecular fragment negative ions via dissociative
1999; Coleman, 2000; Charlton and Humberston, 2001(cid:2). attachment. A dominant mechanism of electron capture
The advent of efficient positron traps marked a turning by molecules is via negative-ion resonant states (cid:1)Chris-
point(cid:1)Surko,Leventhal,etal.,1988;MurphyandSurko, tophorou et al., 1984(cid:2). Dissociative attachment usually
1992; Surko et al., 2005(cid:2), enabling a new generation of proceeds via electron shape resonances of ground or
studies (cid:1)Surko, Passner, et al., 1988; Murphy and Surko, electronically excited molecules. Such resonances are
1991;Iwataetal.,1995;Kurzetal.,1996;Iwata,Greaves, quite common in diatomic, triatomic, and polyatomic
and Surko, 1997(cid:2). Experiments with trapped positrons species at energies in the range (cid:3)0–4 eV. The theoret-
cooled to 300 K permitted studies of test species at low ical description of them involves (cid:1)complex(cid:2) Born-
densities (cid:1)e.g., (cid:3)10−6 amagat(cid:2). This ensured that annihi- Oppenheimerpotential-energysurfaces(cid:1)O’Malley,1966;
lation was strictly due to binary collisions rather than Bardsley, 1968a; Domcke, 1981(cid:2). All the data indicate
many-particle effects (cid:1)Iakubov and Khrapak, 1982(cid:2), and that positrons generally do not form shape resonances
it enabled study of a broader range of chemical species, or electronic Feshbach resonances in low-energy colli-
including low-vapor-pressure liquids and solids. sions with molecules. Instead, energy-resolved annihila-
21amagat=2.69(cid:10)1019cm−3 is the density of an ideal gas at 3ThetheoreticalmaximumforthemagnitudeofZ isgiven
eff
standard temperature and pressure, 273.15K and 101.3kPa, bytheunitaritylimitoftheinelasticcrosssection(cid:1)Landauand
respectively. Lifshitz,1977(cid:2),Z (cid:4)107 forroom-temperaturepositrons.
eff
Rev.Mod.Phys.,Vol.82,No.3,July–September2010
2560 Gribakin,Young,andSurko: Positron-moleculeinteractions:Resonant…
TABLE I. Annihilation rates Z and binding energies (cid:5) for
eff b
selectedmolecules.
(cid:5)
b
Class Molecule Z (cid:1)meV(cid:2)a Z a
eff
Smallinorganics H O 10 (cid:6)0 170b
2
NH 9 (cid:13)0 300b
3
Methylhalides CH F 18 (cid:13)0 250b
3
CH Br 44 40 2000b
3
Alkanes CH 10 (cid:6)0 70b
4
C H 18 (cid:13)0 900c
2 6
C H 26 10 10500c
3 8
C H 50 80 184000c
6 14
C H 98 220 9800000c
12 26
Alcohols CH OH 18 (cid:13)0 750b
3
C H OH 26 45 4500b
2 5
Aromatics C H 42 150 47000c
6 6
C H 68 300 1240000c
10 8
aValues from energy-resolved measurements (cid:1)Young and
FIG. 1. The normalized annihilation rate Zeff(cid:1)(cid:5)(cid:2) for butane Surko, 2008b, 2008c(cid:2); typical uncertainties in Zeff and (cid:5)b are
C H (cid:1)(cid:1)(cid:2)asafunctionofthetotalincidentpositronenergy(cid:5): ±20% and±10meV,respectively.
4 10
(cid:1)a(cid:2)uptothePsformationthreshold,E =3.8eV,and(cid:1)b(cid:2)inthe bMaximumvaluesforpositronenergies(cid:5)(cid:14)50meV.
th
region of the molecular vibrations; dotted curve, the infrared cValuesofZeffattheC-Hresonancepeak.
absorption spectrum (cid:1)Linstrom and Mallard, 2005(cid:2) (cid:1)logarith-
mic vertical scale, arbitrary units(cid:2); solid curve, the vibrational
resonances at lower energies are due to C-C modes and
mode density (cid:1)in arbitrary units(cid:2), with the modes represented
C-H bend modes and exhibit the same downshift.
by Lorentzians with an arbitrary FWHM of 10meV; dashed
There are a number of important chemical trends as-
line,meanenergyoftheC-Hstretchfundamentals.
sociated with resonant annihilation on molecules (cid:1)Mur-
phy and Surko, 1991; Iwata et al., 1995; Young and
tion studies point to the important role played by the Surko, 2008b, 2008c(cid:2). Examples are shown in Table I.
VFR. Very small molecules, such as CO , CH , or H O, have
2 4 2
These vibrational (cid:1)or “nuclear-excited”(cid:2) Feshbach relativelysmallvaluesofZ (cid:1)e.g.,Z /Z(cid:4)10(cid:2),andtypi-
eff eff
resonances involve coupling of the electronic and the cally they do not exhibit resonant annihilation peaks.
nuclearmotionbeyondtheBorn-Oppenheimerapproxi- Positrons either do not bind to these species (cid:1)i.e., (cid:5)
b
mation. It cannot be described by potential-energy sur- (cid:6)0(cid:2) or they bind extremely weakly. With the exception
faces. This type of resonances was originally introduced ofmethane,allofthealkanesexhibitVFRs,withvalues
by Bardsley (cid:1)1968b(cid:2) as an “indirect” mechanism for dis- of (cid:5) increasing linearly with the number of carbon at-
b
sociative electron recombination and described using oms n, and the magnitudes of Z increasing approxi-
eff
Breit-Wigner theory. In the case of electrons, these
matelyexponentiallywithn.Mosthydrocarbons,includ-
VFRsleadtolargeattachmentcrosssectionswhichtypi-
ing aromatic molecules, alkenes, and alcohols, exhibit
cally reach their maximum values at thermal electron
similar resonant annihilation spectra.
energies (cid:1)Christophorou et al., 1984(cid:2). They are also re-
Much progress has been made in the theoretical un-
sponsiblefortheformationoflong-livedparentnegative
derstanding of resonant positron annihilation in mol-
ions for many complex polyatomic molecules. ecules(cid:1)Gribakin,2000,2001;GribakinandGill,2004(cid:2).A
Referring to Fig. 1, the energy of the VFR corre-
quantitative theory has been developed for the case of
sponding to mode (cid:11)is given by energy conservation, isolatedresonancesofIR-activevibrationalmodes,such
(cid:5)(cid:11)=(cid:12)(cid:11)−(cid:5)b, (cid:1)3(cid:2) asthoseobservedinexperimentsforselectedsmallmol-
ecules. The prototypical example is that of the methyl
where(cid:5)bisthepositron-moleculebindingenergyand(cid:12)(cid:11) halides, CH3X, where X is a F, Cl, or Br atom. Positron
isthevibrationalmodeenergy.Thepositronbindingen- coupling to the IR-active modes is evaluated in the di-
ergy (cid:1)i.e., the positron affinity(cid:2) can be measured by the poleapproximationusingdatafromIRabsorptionmea-
downshift of the resonances from the energies of the surements. The only free parameter in the theory is the
vibrational modes. In Fig. 1 for butane, this is most eas- positron binding energy, which can be taken from ex-
ily seen as the shift in the C-H stretch vibrational reso- periment.Thisyieldstheoreticalannihilationspectrafor
nance. The corresponding peak in Z occurs at methylhalidesthatareingoodagreementwiththemea-
eff
330 meV, as compared with the vibrational mode fre- surements (cid:1)Gribakin and Lee, 2006a(cid:2).
quency of 365 meV, indicating that (cid:5) =35 meV. The A more stringent test of the theory relies on the fact
b
Rev.Mod.Phys.,Vol.82,No.3,July–September2010
Gribakin,Young,andSurko: Positron-moleculeinteractions:Resonant… 2561
that positron binding energies are expected to change couplings, and this has not yet been done.
littleuponisotopesubstitution.Fordeuterationthiswas Theenergy-resolvedannihilationexperimentsprovide
confirmed experimentally. The binding energies mea- measures of positron-molecule binding energies, either
sured for CH Cl and CH Br were used to predict Z directly using Eq. (cid:1)3(cid:2) or, for very weakly bound states,
3 3 eff
for their deuterated analogs. The result is excellent indirectly through the dependence of Z on g. To date,
eff
agreement between theory and experiment with no ad- bindingenergiesforabout30 moleculeshavebeenmea-
justable parameters (cid:1)Young et al., 2008(cid:2). In other small sured. They range from (cid:3)1 meV in small molecules,
molecules, such as ethylene and methanol, IR-inactive such as CH F, to (cid:3)300 meV for large alkanes (cid:1)Young
3
modes and multimode vibrations are prominent and and Surko, 2008b, 2008c(cid:2). A recent analysis indicates
must be included to explain the observations (cid:1)Young et that these binding energies increase approximately lin-
al., 2008(cid:2). earlywiththemoleculardipolepolarizabilityanddipole
ThistheoreticalapproachexplainsZ forsmallpoly- moment and, for aromatic molecules, the number of (cid:8)
eff
atomics in which the positron coupling to the mode- bonds (cid:1)Danielson et al., 2009(cid:2).
based VFR and, possibly a few overtones, can be esti- For atoms, comparatively accurate positron binding
mated (cid:1)e.g., when they have dipole coupling(cid:2). Their Z energieshavebeenpredictedtheoreticallyforaboutten
eff
values are between a few hundred and a few thousand.4 species (cid:1)Mitroy et al., 2002(cid:2), but there are no measure-
However, larger molecules with more than one or two ments. There have been a number of calculations for
carbons have values of Z that cannot be explained by positron binding to molecules (cid:1)Schrader and Wang,
eff
this theory (cid:1)cf. Fig. 1 for butane(cid:2). The current physical 1976; Kurtz and Jordan, 1978, 1981; Danby and Tenny-
picture ascribes their large annihilation rates to large son,1988;Bressaninietal.,1998;Strasburger,1999,2004;
densities of vibrational resonances, known as “dark Schrader and Moxom, 2001; Tachikawa et al., 2003;
states” (cid:1)Gribakin, 2000, 2001(cid:2), that are not coupled di- Buenker et al., 2005; Chojnacki and Strasburger, 2006;
rectly to the positron continuum. The positron first at- Gianturco et al., 2006; Buenker and Liebermann, 2008;
tachestothemoleculeviaavibrational“doorwaystate” Carey et al., 2008(cid:2). Most of these molecules have large
(cid:1)e.g., a dipole-allowed mode-based VFR(cid:2) (cid:1)Gribakin and dipole moments which facilitate binding. In contrast,
Gill,2004(cid:2).Thevibrationalenergyisthentransferredto most molecules for which the binding energies are
the dark states in a process known as intramolecular known from experiment are either nonpolar or only
vibrationalenergyredistribution(cid:1)IVR(cid:2).SuchIVRisim- weakly polar. Thus at present, there are almost no spe-
portant for many physical and chemical processes in cies for which experiment and theory can be compared,
molecules, including dissociative attachment (cid:1)Uzer and and so this is a critical area for future research.
Miller, 1991; Nesbitt and Field, 1996(cid:2). Presented here is a review of theoretical and experi-
The magnitudes of resonant contributions to Z ex- mental results for positron annihilation on molecules in
eff
hibit a relatively weak dependence on (cid:5) and on the the range of energies below the positronium formation
incidentpositronenergy(cid:5).Itisoftheformbg=(cid:4)(cid:5) /(cid:5)and threshold. Emphasis is placed upon the case in which
b positrons bind to the target and annihilation proceeds
follows from rather general theoretical considerations.
via the formation of vibrational Feshbach resonances.
Whenthisdependenceisfactoredout,itisfoundexperi-
mentallythattheresultingquantityZ /gscalesas(cid:3)N4, Current knowledge of positron-molecule binding ener-
eff gies,obtainedfrombothexperimentandtheoreticalcal-
where N is the number of atoms in the molecule. This
culations, is summarized. These results are related to
dependenceonN isthoughttoreflecttherapidincrease
studies of positron-induced fragmentation of molecules,
inthedensityofthemolecularvibrationalspectrumwith
annihilation gamma-ray spectra, annihilation in dense
the number of vibrational modes. This dependence is
gases where nonlinear effects are observed, and to
interpreted as evidence that IVR does indeed play an
analogouselectroninteractionswithmoleculesandclus-
important role in the annihilation process.
ters.
Estimates of Z in large molecules, which assume
eff
that the IVR process is complete and all modes are
populatedstatistically,predictZ valuesfarinexcessof II.THEORY
eff
those that are observed. Such estimates also fail to re-
A.Annihilationbasics
produce the energy dependence of Z , which is largely
eff
determined by the mode-based vibrational doorways.
The process of electron-positron annihilation is de-
Onehypothesis,asyetunconfirmed,isthattheIVRpro- scribedbyquantumelectrodynamics(cid:1)QED(cid:2).Inthenon-
cessdoesnotrunto“completion.”Itappearsthatselec-
relativisticBornapproximation,thecrosssectionforan-
tivecouplingofmultimodevibrationsleavesalargepor-
nihilation into two photons averaged over the electron
tion of them inactive. The calculation of Zeff then and positron spins is (cid:1)Berestetskii et al., 1982(cid:2)
requires a detailed knowledge of the vibrational mode
(cid:15)¯2(cid:16)=(cid:8)r02c/v, (cid:1)4(cid:2)
4The heights of resonant peaks in the measured Z spectra wherev istherelativevelocityofthetwoparticles.This
eff
are related to the energy spread (cid:17)(cid:5) of the incident positron cross section obeys the 1/v threshold law which de-
beam. The value of Z (cid:3)103 corresponds to the typical (cid:17)(cid:5) scribes inelastic collisions with fast particles in the final
eff
(cid:3)40meVusedtodate(cid:1)seeSec.III(cid:2). state (cid:1)Landau and Lifshitz, 1977(cid:2).
Rev.Mod.Phys.,Vol.82,No.3,July–September2010
2562 Gribakin,Young,andSurko: Positron-moleculeinteractions:Resonant…
The two-photon annihilation described by Eq. (cid:1)4(cid:2) is attracted to the target by a long-range polarization po-
allowed only when the total spin S of the electron- tential−(cid:18)/2r4,where(cid:18) isthetargetdipolepolarizabil-
d d
positronpairiszero.ForS=1thesmallestpossiblenum- ity, which enhances Z . There is also a short-range en-
eff
ber of annihilation photons is three. The corresponding hancement of Z due to the Coulomb interaction
eff
spin-averaged cross section is (cid:1)Berestetskii et al., 1982(cid:2) between the annihilating electron and positron, which
has the same origin as the expression in Eq. (cid:1)6(cid:2). Finally,
(cid:15)¯3(cid:16)= 34(cid:1)(cid:8)2−9(cid:2)(cid:18)r02c/v, (cid:1)5(cid:2) if the target binds the positron, the annihilation can be
where (cid:18)=e2/(cid:19)c (cid:1)in cgs units(cid:2) is the fine structure con- enhanced by positron capture into this bound state. The
pstoasnitt,ro(cid:18)n(cid:5)a1n/n1i3h7il.aStiionncein(cid:15)¯3m(cid:16)aisny4-0e0letcitmroens ssmysatlelmersthisadno(cid:15)¯m2(cid:16)i-, cprhoostsosne(cid:2)ctiisonsmfoarll,randaimateivlye,c(cid:15)acp(cid:3)tu(cid:18)re3a(cid:1)02i.e(cid:1)B.,ebryesetmetissksiiionetoafl.a,
nated by the two-gamma process. 1982(cid:2). In collisions with molecules, the positron can
Numerically, the cross section in Eq. (cid:1)4(cid:2) is (cid:15)¯2(cid:16) transfer its energy to vibrations forming a positron-
(cid:3)10−8 c/v a.u.5 Hence the annihilation rate is usually moleculecomplex.Thisprocessiseffectiveinenhancing
the annihilation rate. It is the principal focus of the
much smaller than the rates for other atomic collision
processes, even at low positron velocities (cid:1)e.g., thermal, present review.
v(cid:3)0.05 a.u.at300 K(cid:2).Whenafastpositron,suchasthat As follows from its definition by Eq. (cid:1)7(cid:2), Zeff is equal
emitted in a (cid:20)+ decay, moves through matter, it loses to the electron density at the positron,
energy quickly through collisions, first by direct ioniza-
(cid:8)
tion, positronium formation and electronic excitation, Z
(cid:9)
and then by vibrational excitation and elastic collisions. Zeff= (cid:17)(cid:1)r−ri(cid:2)(cid:6)(cid:23)k(cid:1)r1,...,rZ,r(cid:2)(cid:6)2dr1¯drZdr,
As a result, the positrons are typically slow to thermal i=1
energies (cid:1)i.e., (cid:3)25 meV for T=300 K(cid:2) before annihila- (cid:1)8(cid:2)
tion.
At small velocities, v(cid:4)1 a.u., Eq. (cid:1)4(cid:2) must be modi-
where (cid:23) (cid:1)r ,...,r ,r(cid:2) is the total wave function of the
fied to take into account the Coulomb interaction be- k 1 Z
system,withelectroncoordinatesr andpositroncoordi-
tween the electron and positron. The typical momenta i
exchanged in the annihilation process are p(cid:3)mc. The nater.Thiswavefunctiondescribesthescatteringofthe
corresponding separation r(cid:3)(cid:19)/mc is small compared to positron with initial momentum k by the atomic or mo-
lecular target and is normalized to the positron plane
a ,andinthenonrelativisticlimit,theannihilationtakes
0 wave at large separations,
place when the electron and positron are at the same
point. The cross section in Eq. (cid:1)4(cid:2) must then be multi-
plied by the probability density at the origin (cid:1)Landau (cid:23) (cid:1)r ,...,r ,r(cid:2)(cid:7)(cid:24) (cid:1)r ,...,r (cid:2)eik·r, (cid:1)9(cid:2)
k 1 Z 0 1 Z
and Lifshitz, 1977(cid:2),
2(cid:8) where(cid:24) isthewavefunctionoftheinitial(cid:1)e.g.,ground(cid:2)
(cid:6)(cid:21)(cid:1)0(cid:2)(cid:6)2= v(cid:1)1−e−2(cid:8)/v(cid:2), (cid:1)6(cid:2) state of t0he target. For molecules, both (cid:23)k and (cid:24)0 also
depend on the nuclear coordinates, which must be inte-
wherethewavefunction(cid:21)isnormalizedby(cid:21)(cid:1)r(cid:2)(cid:7)eik·rat grated over in Eq. (cid:1)8(cid:2).
r(cid:22)a . This increases the annihilation cross section. Equations (cid:1)7(cid:2) and (cid:1)8(cid:2) determine the annihilation rate
0
The annihilation cross section for many-electron tar- in binary positron-molecule collisions,
gets is traditionally written as (cid:1)Pomeranchuk, 1949;
Fraser, 1968(cid:2)
(cid:7)=(cid:15)vn=(cid:8)r2cZ n, (cid:1)10(cid:2)
a 0 eff
c
(cid:15)a=(cid:15)¯2(cid:16)Zeff=(cid:8)r02vZeff, (cid:1)7(cid:2)
where n is the gas number density. To compare with
where Z represents the effective number of electrons experiment, this rate is averaged over the positron en-
eff
that contribute to the annihilation. In the Born approxi- ergy distribution. For thermal positrons this distribution
is a Maxwellian, while in beam experiments, it is deter-
mation, Z =Z, the total number of target electrons.
eff
At small positron energies (cid:1)e.g., (cid:5)(cid:4)1 eV(cid:2), however, mined by the parameters of the beam. Empirically Eq.
(cid:1)10(cid:2) is also used to describe experiments at high densi-
Z can be different from Z. First, there is a strong re-
eff ties where Z becomes density dependent (cid:1)see Sec.
pulsionbetweenthepositronandtheatomicnuclei.This eff
VIII.C(cid:2).
preventsthepositronfrompenetratingdeepintotheat-
oms so that the annihilation involves predominantly
electrons in the valence and near-valence subshells,
therebyreducingZ .Ontheotherhand,thepositronis
eff
B.Gamma-rayspectraandannihilationrates
5We make use of atomic units (cid:1)a.u.(cid:2), in which m=(cid:6)e(cid:6)=(cid:19)=1, In the nonrelativistic limit, the two-photon QED an-
c=(cid:18)−1(cid:5)137a.u.,andtheBohrradiusa =(cid:19)2/me2(cid:1)incgsunits(cid:2) nihilation amplitude can be expressed in terms of an
0
alsoequalsunity. effective annihilation operator
Rev.Mod.Phys.,Vol.82,No.3,July–September2010
Gribakin,Young,andSurko: Positron-moleculeinteractions:Resonant… 2563
(cid:8)
tions of different final states. For example, in partially
Oˆ (cid:1)P(cid:2)(cid:10) e−iP·r(cid:21)ˆ(cid:1)r(cid:2)(cid:25)ˆ(cid:1)r(cid:2)dr, (cid:1)11(cid:2) fluorinated alkanes, annihilation with the tightly bound
a
fluorine 2p electrons results in a broader spectral com-
where (cid:21)ˆ(cid:1)r(cid:2) and (cid:25)ˆ(cid:1)r(cid:2) are the electron and positron de- ponent than annihilation with the more diffuse C-H
structionoperators6andP isthetotalmomentumofthe bond electrons. This allows one to deduce the relative
photons(cid:1)Ferrell,1956;Lee,1957;DunlopandGribakin, fraction of the corresponding annihilation events (cid:1)Iwata
2006(cid:2). The probability distribution of P in an annihila- et al., 1997(cid:2); see Sec. VIII.A.
Thetotalannihilationrateinthestatei leadingtothe
tion event is given by
finalstatefisobtainedbyintegrationoverthemomenta
W(cid:1)P(cid:2)=(cid:8)r2c(cid:11)(cid:12)f(cid:6)Oˆ (cid:1)P(cid:2)(cid:6)i(cid:13)(cid:11)2, (cid:1)12(cid:2) (cid:8)
f 0 a d3P
(cid:7) =(cid:8)r2c (cid:11)(cid:12)f(cid:6)Oˆ (cid:1)P(cid:2)(cid:6)i(cid:13)(cid:11)2 , (cid:1)16(cid:2)
where (cid:6)i(cid:13) is the initial state with Z electrons and the f 0 a (cid:1)2(cid:8)(cid:2)3
positron (cid:1)e.g., that with the wave function (cid:23) (cid:2) and (cid:6)f(cid:13) is
k and the total annihilation rate in state i is
the state of Z−1 electrons after the annihilation.
(cid:8)
For P=0, the two photons are emitted in opposite di- (cid:9)
rections and have equal energies, E(cid:16)1=E(cid:16)2(cid:10)E(cid:16)(cid:5)mc2. (cid:7)= (cid:7)f=(cid:8)r02c (cid:12)i(cid:6)nˆ−(cid:1)r(cid:2)nˆ+(cid:1)r(cid:2)(cid:6)i(cid:13)dr, (cid:1)17(cid:2)
For P(cid:3)0 the photon energy is Doppler shifted, e.g., f
E(cid:16)1=E(cid:16)+mc(cid:6)V(cid:6)cos(cid:26), (cid:1)13(cid:2) where nˆ−(cid:1)r(cid:2)=(cid:21)ˆ†(cid:1)r(cid:2)(cid:21)ˆ(cid:1)r(cid:2) and nˆ+(cid:1)r(cid:2)=(cid:25)ˆ†(cid:1)r(cid:2)(cid:25)ˆ(cid:1)r(cid:2) are the elec-
tron and positron density operators. The annihilation
where V=P/2m is the center-of-mass velocity of the rate is thus given by the expectation value of the elec-
electron-positron pair and (cid:26)is the angle between V and trondensityatthepositron.Equation(cid:1)17(cid:2)givesthetwo-
the direction of the photon. Averaging the distribution photonannihilationrateinasystemofonepositronand
of the Doppler shifts (cid:27)=E(cid:16)1−E(cid:16)=(cid:1)Pc/2(cid:2)cos(cid:26), over the one target atom or molecule. For a positron moving
direction of emission of the photons, gives the photon through a gas of density n, the annihilation rate takes
energy spectrum the form of Eq. (cid:1)10(cid:2). Normalizing the initial state i to
(cid:8)(cid:8)
1 (cid:28) PdPd(cid:29) one positron per unit volume, as (cid:23)k in Eq. (cid:1)9(cid:2), one ob-
w(cid:1)(cid:27)(cid:2)= W(cid:1)P(cid:2) P. (cid:1)14(cid:2) tains
f c f (cid:1)2(cid:8)(cid:2)3 (cid:8)
2(cid:6)(cid:27)(cid:6)/c
In Cartesian coordinates, Zeff= (cid:12)i(cid:6)nˆ−(cid:1)r(cid:2)nˆ+(cid:1)r(cid:2)(cid:6)i(cid:13)dr. (cid:1)18(cid:2)
(cid:8)(cid:8)
w(cid:1)(cid:27)(cid:2)= 2 W(cid:1)P ,P ,2(cid:27)/c(cid:2)dPxdPy. (cid:1)15(cid:2) In the coordinate representation, this yields Eq. (cid:1)8(cid:2).
f c f x y (cid:1)2(cid:8)(cid:2)3 In the independent-particle approximation, the elec-
tronic parts of the initial and final states are Slater de-
This form shows that the energy spectrum is propor-
terminants constructed from the electron orbitals (cid:1)e.g.,
tional to the probability density for a component of P.
in the Hartree-Fock scheme(cid:2). The incident positron is
This quantity can be measured either by sampling the
described by its own wave function (cid:25)(cid:1)r(cid:2), and the anni-
Doppler spectrum of the gamma rays or by measuring k
theangulardeviationofthetwophotons(cid:1)seeSec.III.F(cid:2). hilation amplitude (cid:12)f(cid:6)Oˆa(cid:1)P(cid:2)(cid:6)i(cid:13) takes the form
(cid:8)
When a low-energy positron annihilates with a bound
electron with energy (cid:5)n, the mean photon energy E(cid:16)is Ank(cid:1)P(cid:2)= e−iP·r(cid:21)n(cid:1)r(cid:2)(cid:25)k(cid:1)r(cid:2)dr, (cid:1)19(cid:2)
shiftedby(cid:5) /2relativetomc2.Thisshiftismuchsmaller
n
thanthetypicalDopplershift(cid:27)duetothemomentumof where (cid:21)(cid:1)r(cid:2) is the orbital of the annihilated electron. In
(cid:4) n
the bound electron P(cid:3) 2m(cid:6)(cid:5) (cid:6), which corresponds to this approximation
(cid:4) n
(cid:27)(cid:3)Pc(cid:3) (cid:6)(cid:5) (cid:6)mc2(cid:22)(cid:6)(cid:5) (cid:6).Theresultingwidthandshapeof (cid:8)
n n Z
the gamma spectrum contain important information Z =(cid:9) (cid:6)(cid:21)(cid:1)r(cid:2)(cid:6)2(cid:6)(cid:25)(cid:1)r(cid:2)(cid:6)2dr, (cid:1)20(cid:2)
eff n k
about the bound electrons.
n=1
Inmostexperiments,theannihilationphotonsarenot
i.e., the average product of the electron and positron
detected in coincidence with the final state f, and the
densities.
observed spectrum is the sum over all final states w(cid:1)(cid:27)(cid:2)
=(cid:9)w(cid:1)(cid:27)(cid:2). However, this spectrum still reveals contribu-
f f
C.Positron-moleculewavefunction
6In Eq. (cid:1)11(cid:2) the spin indices in (cid:21)ˆ(cid:1)r(cid:2) and (cid:25)ˆ(cid:1)r(cid:2) are suppressed, Thewavefunction(cid:23) forthepositroncollidingwitha
k
and summation over them is assumed. This form can be used molecule can be written as (cid:1)Gribakin, 2000, 2001(cid:2)
in systems with paired electron spins or when averaging over
tthipelipeodsbityrotnhespsipni.nT-ahveemraogdedulQusE-sDqufaarcetdoram(cid:8)rp02lcit.uIdnegiesnthereanl,mounle- (cid:23)k=(cid:23)k(cid:1)0(cid:2)+(cid:9)(cid:11) (cid:5)(cid:23)−(cid:11)(cid:12)(cid:5)(cid:23)(cid:11)(cid:11)+(cid:6)V(cid:1)i(cid:6)/(cid:23)2(cid:2)k(cid:1)0(cid:30)(cid:2)(cid:13)(cid:11). (cid:1)21(cid:2)
shouldusethespin-singletcombinationoftheannihilationop-
(cid:4)
eratorsinEq.(cid:1)11(cid:2),(cid:1)1/ 2(cid:2)(cid:1)(cid:21)ˆ↑(cid:25)ˆ↓−(cid:21)ˆ↓(cid:25)ˆ↑(cid:2),togetherwiththetwo- The first term on the right-hand side describes direct or
photonannihilationfactor4(cid:8)r2c. potential scattering of the positron by the target. The
0
Rev.Mod.Phys.,Vol.82,No.3,July–September2010
2564 Gribakin,Young,andSurko: Positron-moleculeinteractions:Resonant…
corresponding wave function (cid:23)(cid:1)0(cid:2) is determined by the To calculate Z , the wave function from Eq. (cid:1)21(cid:2) is
k eff
positron interaction with the charge distribution of the substituted into Eq. (cid:1)8(cid:2), which yields
ground-statetargetandelectron-positroncorrelationef-
Z
fects (cid:1)e.g., target polarization and virtual Ps formation(cid:2). Z =(cid:12)(cid:23) (cid:6)(cid:9)(cid:17)(cid:1)r−r(cid:2)(cid:6)(cid:23) (cid:13)
eff k i k
ItneglectsthecouplingV betweentheelectron-positron i=1 (cid:14) (cid:15)
and nuclear (cid:1)vibrational(cid:2) degrees of freedom. The sec-
Z
ond term describes positron capture into the vibrational =(cid:12)(cid:23)(cid:1)0(cid:2)(cid:6)(cid:9)(cid:17)(cid:1)r−r(cid:2)(cid:6)(cid:23)(cid:1)0(cid:2)(cid:13)+ interference
Feshbachresonances.Itispresentformoleculesthatcan k i k terms
i=1
bind the positron. These resonances correspond to vi-
cobocramcutiprolnewxa,lhlyeenmexbtcheideteddepdosstiiatntreostnh(cid:23)ee(cid:11)npeoorsfgitytrhoe(cid:5)n=pckoo2sn/itt2rinoiunsu-mmclo.olseTechuetloye + 2(cid:8)2(cid:9) A *(cid:12)(cid:23) (cid:6)(cid:9)i=Z1(cid:17)(cid:1)r−ri(cid:2)(cid:6)(cid:23)(cid:11)(cid:13)A(cid:11) , (cid:1)25(cid:2)
(cid:5)(cid:11)=E(cid:11)−(cid:5)b, where (cid:5)b is the positron binding energy and k (cid:11)(cid:1)(cid:5)−(cid:5) − 2i(cid:30) (cid:2)(cid:1)(cid:5)−(cid:5)(cid:11)+ 2i(cid:30)(cid:11)(cid:2)
E(cid:11)is the vibrational excitation energy of the positron- where the capture amplitude A(cid:11)is related to the elastic
molecule complex. Equation (cid:1)21(cid:2) has the appearance width by (cid:30)(cid:11)e=2(cid:8)(cid:6)A(cid:11)(cid:6)2. The terms on the right-hand side
ofastandardperturbation-theoryformula,buttheener-
describethecontributionsofdirectandresonantannihi-
gies of the positron-molecule quasibound states (cid:23)(cid:11) lation and the interference between the two. We now
in the denominator are complex, (cid:5)(cid:11)−(cid:1)i/2(cid:2)(cid:30)(cid:11), where examine the two main contributions in detail.
(cid:30)(cid:11)=(cid:30)(cid:11)a+(cid:30)(cid:11)e+(cid:30)(cid:11)i is the total width of the resonance. In The separation of the wave function into the direct
atomicunits(cid:30)(cid:11)isequaltothedecayrateoftheresonant and resonant parts in Eq. (cid:1)21(cid:2) is valid because the posi-
state. It contains contributions of positron annihilation tron VFRs are narrow. This is a consequence of the
and elastic escape (cid:30)(cid:11)a and (cid:30)(cid:11)e, respectively, and possibly weakness of coupling between the positron and the vi-
also the inelastic escape rate (cid:30)(cid:11)i. The latter describes brational motion (cid:1)i.e., small capture widths (cid:30)(cid:11)e, see Sec.
positron autodetachment accompanied by vibrational II.F(cid:2). In spite of this, the resonant contribution to the
transitions to the states other than the initial state. annihilation rate for complex polyatomics exceeds the
Molecular rotations are, in general, not expected to direct contributions by orders of magnitude.
affect positron annihilation. The rotational motion is
slow compared to the motion of the positron or the vi-
D.Directannihilation:Virtualandweaklyboundpositron
brational motion. Accordingly, direct scattering can be
states
considered for fixed molecular orientation and the re-
sults averaged over the orientations. Positron capture in The potential scattering wave function (cid:23)(cid:1)0(cid:2) satisfies
VFRs at low energies is dominated by the s wave or at k
the Schrödinger equation
most a few lower partial waves. Hence in the capture
process,theangularmomentumofthemoleculeremains (cid:1)T+U−E (cid:2)(cid:23)(cid:1)0(cid:2)=(cid:5)(cid:23)(cid:1)0(cid:2), (cid:1)26(cid:2)
0 k k
unchanged or changes little.
where T is the kinetic energy operator for the electrons
The positron capture amplitude (cid:12)(cid:23)(cid:11)(cid:6)V(cid:6)(cid:23)k(cid:1)0(cid:2)(cid:13) deter- and positron, U is the sum of all Coulomb interactions
mines the elastic width in state (cid:11), between the particles (cid:1)with the nuclei at their equilib-
(cid:8)
(cid:30)(cid:11)e=2(cid:8) (cid:11)(cid:12)(cid:23)(cid:11)(cid:6)V(cid:6)(cid:23)k(cid:1)0(cid:2)(cid:13)(cid:11)2k(cid:1)2d(cid:8)(cid:29)(cid:2)3k. (cid:1)22(cid:2) riuFmorpopsoistiitornosn(cid:2),eannedrgEie0sisbtehleowtarthgeetPgsr-ofuonrmd-asttiaotneethnreersghy-.
old, annihilation occurs when the positron is within the
If the positron interaction with the vibrations cannot be range of the target ground-state electron cloud. At such
described by perturbation theory, Eqs. (cid:1)21(cid:2) and (cid:1)22(cid:2) re- distances, the interaction U between the particles is
mplaaicnedvablyidthperiorvnidoendpetrhteurabmatpivlietuvdaelsue(cid:12)s(cid:23).(cid:11)(cid:6)V(cid:6)(cid:23)k(cid:1)0(cid:2)(cid:13) are re- m(cid:5)(cid:23)u(cid:1)c0h(cid:2)tgerrematienrEthqa.n(cid:1)2t6h(cid:2)ecpaonsbiteronnegelneecrtgeyd,(cid:5)a.nTdhtehreefsoorleu,titohne
k
According to Eq. (cid:1)17(cid:2), the annihilation rate of the (cid:23)(cid:1)0(cid:2) at these small separations does not depend on (cid:5),
k
positron-molecule state (cid:23)(cid:11)is given by except through a normalization factor.
When the positron is outside the target, (cid:23)(cid:1)0(cid:2) contains
(cid:30)a=(cid:8)r2c(cid:31) , (cid:1)23(cid:2) k
(cid:11) 0 ep contributions of the incident and scattered positron
where(cid:31) istheaverageelectrondensityatthepositron, waves, (cid:16) (cid:17)
ep
(cid:8) eikr
(cid:31)ep= (cid:9)Z (cid:17)(cid:1)r−ri(cid:2)(cid:6)(cid:23)(cid:11)(cid:1)r1,...,rZ,r(cid:2)(cid:6)2dr1¯drZdr, (cid:23)k(cid:1)0(cid:2)(cid:1)r1,...,rZ,r(cid:2)(cid:7)(cid:24)0(cid:1)r1,...,rZ(cid:2) eik·r+fkk(cid:1) r ,
i=1 (cid:1)27(cid:2)
(cid:1)24(cid:2)
where fkk(cid:1) is the scattering amplitude and k(cid:1)=kr/r. In-
with the integration extending to the nuclear coordi- side the target, (cid:23)(cid:1)0(cid:2) is determined by matching it with
k
nates in the wave function (cid:23)(cid:11). The amplitude of the Eq. (cid:1)27(cid:2) at the target boundary r=R, where R is the
nuclear motion is small, and (cid:31) is expected to depend characteristic radius of the target. For small positron
ep
weaklyonthedegreeofvibrationalexcitationinstate(cid:11). momenta kR!1, the scattering is dominated by the s
Rev.Mod.Phys.,Vol.82,No.3,July–September2010
Gribakin,Young,andSurko: Positron-moleculeinteractions:Resonant… 2565
wave, and the amplitude fkk(cid:1) can be replaced by the
s-waveamplitudef .Asaresult,theintegrandinEq.(cid:1)8(cid:2)
0
for Z is proportional to (cid:6)1+f /R(cid:6)2 (cid:1)Dzuba et al., 1993(cid:2). 1000
eff 0
This gives the following estimate for Z due to direct
eff
annihilation (cid:1)Gribakin, 2000(cid:2),
eff 100
Z(cid:1)dir(cid:2)(cid:7)4(cid:8)(cid:31)(cid:17)R(cid:1)R2+2R Re f +(cid:15) /4(cid:8)(cid:2), (cid:1)28(cid:2) Z
eff e 0 el
where(cid:31) istheeffectiveelectrondensityintheregionof 10
e
annihilation,(cid:17)R istherangeofdistanceswherethepos-
itron annihilates, and (cid:15) is the elastic cross section. At
el
small positron energies, (cid:15) (cid:7)4(cid:8)(cid:6)f (cid:6)2, and in the zero- 1 0.01 0.1 1
el 0 Positronmomentum(a.u.)
energylimit(cid:15) =4(cid:8)a2,wherea isthepositronscattering
el
length, a=−f0 at k=0.7 FIG. 2. (cid:1)Color online(cid:2) Comparison of the Z values calcu-
Asimpleestimateofthefactor4(cid:8)(cid:31)e(cid:17)R(cid:10)FinEq.(cid:1)28(cid:2) lated using the SMC method for C2H2 (cid:1)squaefrfes(cid:2) and C2H4
is obtained using the Ps density at the origin, (cid:31)e(cid:3)(cid:31)Ps (cid:1)circles(cid:2) by Varella et al. (cid:1)2002(cid:2) (cid:1)see Sec. II.H.1(cid:2) with the fit
=1/8(cid:8), and (cid:17)R(cid:3)1, which yields F(cid:3)0.5. Equation (cid:1)28(cid:2) usingEq.(cid:1)29(cid:2)withaconstantverticaloffset.Parametersofthe
tthheengsehoomwestrthicaatltchreomssasgencittiuodneooffZthe(cid:1)deffir(cid:2)taisrgceotm(cid:1)pinaraabtolemtioc ficut:rvCe2(cid:2)H,F2=(cid:1)s0o.l2i3d0c,uarnvde(cid:2)",=F0=.003.27621., and "=0.0041; C2H4 (cid:1)dashed
units(cid:2), unless (cid:15) is much greater than R2.
el
When the scattering cross section is large, the annihi-
ofshort-rangecorrelationtermsintheSMCcalculation,
lation rate is greatly enhanced. This occurs when the
which would enhance the electron density at the posi-
positron has a virtual or a bound state close to zero tron (cid:1)see Sec. II.H.1(cid:2).
energy (cid:1)Goldanskii and Sayasov, 1964(cid:2). Such states are
Positron virtual states explain the large thermal Z
characterizedbyasmallparameter"=1/a,(cid:6)"(cid:6)!R−1.Itis eff
values observed at room temperature in heavier noble
related to the energy of the bound state (cid:5)0=−"2/2 (cid:1)for gases (cid:1)Dzuba et al., 1993, 1996(cid:2). The value of Z =400
"(cid:13)0(cid:2) or virtual state (cid:5)0="2/2 (cid:1)for "(cid:6)0(cid:2). This param- observed for Xe (cid:1)Murphy and Surko, 1990(cid:2) is celfofse to
eter determines the low-energy s-wave scattering ampli-
the maximum direct annihilation rate for thermal posi-
tude f0=−(cid:1)"+ik(cid:2)−1 and cross section (cid:15)el(cid:7)4(cid:8)/(cid:1)"2+k2(cid:2) trons at 300 K. It is estimated from Eq. (cid:1)29(cid:2) to be
(cid:1)Landau and Lifshitz, 1977(cid:2). For small ", this cross sec-
tion can be much greater than the geometrical size of Z(cid:1)dir(cid:2)(cid:4)103. (cid:1)30(cid:2)
the target. The last term in brackets in Eq. (cid:1)28(cid:2) then eff
dominates, and Z(cid:1)dir(cid:2) shows a similar enhancement Higher Z values observed in many polyatomics (cid:1)see,
eff eff
(cid:1)Dzuba et al., 1993; Mitroy and Ivanov, 2002(cid:2)8 e.g., Table I(cid:2) can be understood only by considering
positron-molecule binding and resonances.
F
Z(cid:1)dir(cid:2)(cid:7) . (cid:1)29(cid:2) The annihilation rate for the positron bound to an
eff "2+k2 atom or molecule is
The applicability of Eq. (cid:1)29(cid:2) is shown in Fig. 2. It (cid:8)
Z
shows the Z values from the Schwinger multichannel (cid:9)
(cid:1)SMC(cid:2) calcuelfaftion for C H and C H (cid:1)Varella et al., (cid:30)a=(cid:8)r02c (cid:17)(cid:1)r−ri(cid:2)(cid:6)(cid:23)0(cid:1)r1,...,rZ,r(cid:2)(cid:6)2dr1¯drZdr,
2 2 2 4 i=1
2002(cid:2), fitted using Eq. (cid:1)29(cid:2) with a small vertical offset to
(cid:1)31(cid:2)
accountforthenonresonantZ background.According
eff
to the SMC calculation, both molecules possess virtual
where (cid:23) is the wave function of the bound state. For a
positron states. This results in the characteristic rise of 0
Z at small positron momenta described by Eq. (cid:1)29(cid:2). weakly bound state (cid:1)e.g., (cid:5)b!1 eV(cid:2) (cid:30)a can be estimated
Thefef virtual level in C H (cid:1)fitted value "=0.0041(cid:2) lies in a way similar to that used for Ze(cid:1)dffir(cid:2) above. When the
closer to zero energy t2ha2n in C H (cid:1)"=0.0372(cid:2), which positron is outside the target (cid:1)r(cid:13)R(cid:2), (cid:23)0 takes the form
2 4
manifests in the large Z values for acetylene. The fit-
eff A
ted factor F(cid:5)0.25 for the two molecules is smaller than (cid:23) (cid:1)r ,...,r ,r(cid:2)(cid:7)(cid:24) (cid:1)r ,...,r (cid:2) e−"r, (cid:1)32(cid:2)
the estimate obtained from high-quality atomic calcula- 0 1 Z 0 1 Z r
tions (cid:1)see below(cid:2). This is likely an indication of the lack
where A is the asymptotic normalization constant.9 For
weakbinding(cid:1)"!R−1(cid:2)themaincontributiontothenor-
7If the target molecule has a permanent dipole moment , malization integral,
the long-range dipole potential (cid:1)·r/r3 dominates the low-
energy scattering (cid:1)Fabrikant, 1977(cid:2). This makes (cid:15) infinite,
el
whileZ remainsfinite,makingEq.(cid:1)28(cid:2)invalid. 9Equation (cid:1)32(cid:2) assumes that the ionization potential of the
eff
8The long-range polarization potential −(cid:18)/2r4 modifies the atomic system satisfies E(cid:13)E . For E(cid:6)E the asymptotic
d i Ps i Ps
near-threshold form of (cid:15) and Z(cid:1)dir(cid:2) (cid:1)Gribakin, 2000; Mitroy, form is that of Ps bound to the positive ion (cid:1)Mitroy et al.,
el eff
2002(cid:2),butEq.(cid:1)29(cid:2)canstillbeusedasanestimate. 2002(cid:2).
Rev.Mod.Phys.,Vol.82,No.3,July–September2010
2566 Gribakin,Young,andSurko: Positron-moleculeinteractions:Resonant…
can use this value to evaluate the annihilation rates for
LiH
positron-molecule bound states from Eq. (cid:1)35(cid:2), provided
1.5
their binding energies are known.
1)
-s1.0
90 Mg E.Resonantannihilation
(1 Ag
Γ Cu The effect of resonances on Z is described by the
0.5 Be Cd eff
Zn second and third terms in Eq. (cid:1)25(cid:2). It is dominated by
thediagonalpartofthedoublesuminthelastterm.The
0.0 off-diagonalandinterferencetermsvanishuponaverag-
0 0.1 0.2 0.3
κ(a.u.) ing over the positron energy and can usually be ne-
glected. The resonant contribution to the annihilation
FIG.3. (cid:1)Coloronline(cid:2)Dependenceoftheannihilationrate(cid:30)a
(cid:4) cross section is described by the Breit-Wigner formula
for positron bound states on the parameter "= 2(cid:5)b: solid (cid:1)Landau and Lifshitz, 1977(cid:2),
circles,recentresultsforsixatoms(cid:1)BromleyandMitroy,2002,
2s(cid:1)Mu0l0ti6tsr,of2oy0r1at0nh;deMsReiytarzothoyimkehst,a(cid:1)2Ml0.,0it02r(cid:2)0o;0yd2a,est2h0ae0ld.8,(cid:2)l;2in0oe0p2ei(cid:2)snathcniedrcfilLetisH(cid:30),ae=mar5ol.i3lee"rcur(cid:1)eilne- (cid:15)a= k(cid:8)2(cid:9)(cid:11) (cid:1)(cid:5)−b(cid:5)(cid:11)(cid:11)(cid:30)(cid:2)(cid:11)a2(cid:30)+(cid:11)e41(cid:30)(cid:11)2, (cid:1)37(cid:2)
109s−1(cid:2)whichcorrespondstoF=0.66a.u. where b(cid:11)is the degeneracy of the (cid:11)th resonance. Equa-
tions (cid:1)7(cid:2), (cid:1)23(cid:2), and (cid:1)37(cid:2) then give the resonant Z ,
(cid:8) eff
(cid:6)(cid:23)0(cid:1)r1,...,rZ,r(cid:2)(cid:6)2dr1¯drZdr=1, (cid:1)33(cid:2) Ze(cid:1)rfefs(cid:2)= (cid:8)k(cid:31)ep(cid:9)(cid:11) (cid:1)(cid:5)−(cid:5)b(cid:11)(cid:11)(cid:2)(cid:30)2(cid:11)e+ 41(cid:30)(cid:11)2. (cid:1)38(cid:2)
comes from large positron separations where Eq. (cid:1)32(cid:2) is Thecontactdensity(cid:31) canbeestimatedfromEq.(cid:1)36(cid:2)if
ep
valid. This yields thepositronbindingenergyisknown.TocalculateZ(cid:1)res(cid:2),
eff
(cid:4) one also needs the energies and widths of the reso-
A= "/2(cid:8). (cid:1)34(cid:2) nances. The former are determined by the positron
bindingenergyandthevibrationalexcitationenergiesof
By matching the wave function (cid:23)0 in Eq. (cid:1)31(cid:2) at r=R the positron-molecule complex. The elastic and total
with the asymptotic form in Eq. (cid:1)32(cid:2), one obtains rates depend on the strength of coupling between the
positron and the vibrational motion and, for overtones
"
(cid:30)a(cid:7)(cid:8)r2c4(cid:8)(cid:31)(cid:17)R(cid:6)A(cid:6)2=(cid:8)r2cF (cid:1)35(cid:2) and combination excitations, on the strength of anhar-
0 e 0 2(cid:8) monic terms in the vibrational Hamiltonian. This makes
an ab initio calculation of resonant Z a multifaceted
(cid:1)Gribakin, 2001(cid:2). Hence the electron-positron contact eff
problem.
density from Eq. (cid:1)24(cid:2) is estimated by
(cid:31) (cid:7)(cid:1)F/2(cid:8)(cid:2)". (cid:1)36(cid:2) F.Resonancesduetoinfrared-activemodes
ep
Equation (cid:1)35(cid:2) shows that (cid:30)a is proportional to " Onecaseinwhichsuchacalculationispossibleisthat
(cid:4)
= 2(cid:5) (cid:18)i.e.,tothesquarerootofthebindingenergy;see of isolated vibrational resonances of IR-active funda-
b
Mitroy and Ivanov (cid:1)2002(cid:2) for an alternative mentals (cid:1)Gribakin and Lee, 2006a(cid:2). Consider a small
derivation(cid:19).10 This relationship between (cid:30)a and "is con- polyatomic molecule which supports a bound positron
firmed by positron-atom bound-state calculations statewithasmallbindingenergy(cid:5) ="2/2!1.Thewave
b
(cid:1)Mitroyetal.,2002(cid:2).Figure3showsvaluesforsixatoms function of the bound positron is very diffuse. Outside
with E (cid:13)E , namely, Be, Mg, Cd, Cu, Zn, and Ag, ob- the molecule it behaves as (cid:25)(cid:1)r(cid:2)=Ar−1e−"r, with A given
i Ps 0
tained using high-quality configuration interaction and by Eq. (cid:1)34(cid:2).
stochastic variational methods (cid:1)see Sec. II.H.2 for de- Suppose that the vibrational modes in this molecule
tails(cid:2). Note that the datum for the LiH molecule also arenotmixedwithovertonesorcombinationvibrations.
follows this trend, in spite of its large dipole moment Due to the weakness of the positron binding, the vibra-
(cid:18) =5.9 D (cid:1)Lide, 2000(cid:2)(cid:19) and relatively strong binding. A tional excitation energies of the positron-molecule com-
linearfitthroughtheatomicdatapointsgivesavaluefor plex are close to the vibrational fundamentals (cid:12)(cid:11)of the
the factor F=4(cid:8)(cid:31)e(cid:17)R in Eq. (cid:1)35(cid:2), namely, F(cid:5)0.66 a.u., neutral molecule, E(cid:11)(cid:5)(cid:12)(cid:11).11 In this case the sum in Eq.
which is close to the rough estimate given above. One (cid:1)38(cid:2) is over the modes (cid:11), and the resonant energies are
10Equation (cid:1)32(cid:2) is valid if the positron-target interaction is 11There is extensive experimental evidence that E(cid:11)(cid:5)(cid:12)(cid:11)for
short range. It must be modified if the molecule has a dipole most resonances observed (cid:1)cf. Secs. IV and V(cid:2). Apparent ex-
moment(cid:18)see,e.g.,Fabrikant(cid:1)1977(cid:2)(cid:19).However,Eq.(cid:1)35(cid:2)canbe ceptions,whereshifts(cid:3)10–20meVareobservedaretheC-H
used as an estimate if the dipole force does not play a domi- stretch mode of CH F and the O-H stretch in methanol (cid:1)cf.
3
nantroleinthebinding. Sec.IV(cid:2).
Rev.Mod.Phys.,Vol.82,No.3,July–September2010