Positron-Hydrogen Scattering, Annihilation, and Positronium Formation

13 Pages·2016·0.42 MB·English

by  Anand K. Bhatia

atoms
Article
Positron-Hydrogen Scattering, Annihilation,
and Positronium Formation
AnandK.Bhatia
HeliophysicsScienceDivision,NASA/GoddardSpaceFlightCenter,Greenbelt,MD20771,USA;
[email protected]
AcademicEditor:ChiYuHu
Received:14September2016;Accepted:26October2016;Published:4November2016
Abstract: Inpreviouspapers(BhatiaA.K.2007,2012)ahybridtheoryforthescatteringofelectrons
fromahydrogenicsystemwasdevelopedandappliedtocalculatescatteringphaseshifts,Feshbach
resonances,andphotoabsorptionprocesses. Thisapproachisnowbeingappliedtothescattering
of positrons from hydrogen atoms. Very accurate phase shifts, using the Feshbach projection
operator formalism, were calculated previously (Bhatia A.K. et al. 1971 and Bhatia et al. 1974a).
The present results, obtained using shorter expansions in the correlation function, along with
long-range correlations in the Schrödinger equation, agree very well with the results obtained
earlier. Thescatteringlengthisalsocalculatedandthepresentresultsarecomparedwiththeprevious
results. Annihilation cross-sections, and positronium formation cross-sections, calculated in the
distorted-waveapproximation,arealsopresented.
Keywords: positronscattering;annihilation;positronformation
1. Introduction
Scatteringbyhydrogenicsystemshasbeencarriedoutusingvariousapproximations. Among
themisthemethodofpolarizedorbitalsofTemkin(1959)[1],whichtakesintoaccountthedistortion
producedinthetargetbytheincidentparticleintheansatzforthewavefunctionforthescattering
process. However,thismethodisnotvariationalanddoesnotprovideanyboundsonthecalculated
phaseshifts. Analternateapproachistointroduceseparatecorrelationfunctionsandamalgamate
themintothescatteringproblemviaanopticalpotentialinthescatteringequation. Themostrigorous
approachusestheprojectionoperatorformalismofFeshbach[2](1962). Thisapproachwasapplied
tocalculateaccurateSandPphaseshiftsforelectron-hydrogenandelectron-heliumionscattering
byBhatiaandTemkin(2001)[3]andBhatia(2002,2004,and2006)[4–6]. Thephaseshiftsobtained
havearigorouslowerboundtotheexactphaseshifts. Theclose-couplingapproachalsoprovides
rigorouslowerbounds,whiletheKohn-variationalprinciple,andothermethodsrelatedtoit,donot.
AsindicatedintheformalismofthehybridtheorybyBhatia(2007,2008)[7,8]itisnotnecessaryto
useFeshbachprojectionoperatorstocalculatetheopticalpotentialwhichreplacesthemany-particle
Schrödinger equation with a single-particle equation. The phase shifts calculated by Bhatia (2007,
2008)[7,8]haverigorouslowerbounds. Wenowapplythisapproachtothescatteringofpositrons
fromhydrogenatoms. Therearenononlocalpotentialsinthescatteringequationforthescattering
function, as in the electron scattering, because of the absence of exchange between positrons and
electrons. Previously,Feshbachprojectionoperatorswereusedtocarryoutthepositron-hydrogen
scattering calculations of Bhatia et al (1971, 1974) [9,10] and the correction due to the effect of the
targetpolarizationwasappliedtotheextrapolatedvariationalresults. Inspiteofthefactthatthis
adhoccorrectiondestroyedthevariationalbound,thefinalresultsobtainedbyBhatiaetal. (1971,
1974a) [9,10] have been considered accurate and have stood the test of time. We now apply the
hybridtheorytopositronscatteringandthepresentcalculationincludesshort-rangeandlong-range
Atoms2016,4,27;doi:10.3390/atoms4040027 www.mdpi.com/journal/atoms
Atoms2016,4,27 2of13
correlations at the same time, as mentioned above. The details of this formalism have been given
inpreviouspublicationsofBhatia(2007, 2008)[7,8]andtherefore, webrieflydescribethemethod.
Thewavefunctionisgivenby:
Ψ (→r ,→r ) = uL(r1)Y ((cid:95)r )Φpol(r ,r )+ ∑N C Φλ(→r ,→r ). (1)
L 1 2 r L0 1 1 2 λ L 1 2
1 λ=1
→ (cid:42)
where the positron coordinate is given by r , the electron coordinate by r , and the summation
1 2
overλisfrom1toN,thenumberoftermsintheexpansion;C areunknowncoefficients,andΦλ is
λ L
→ →
thecorrelationfunction. Thecorrelationfunctionsareshort-rangeinnaturein r and r ,andare
1 2
normalizedtounity. TheirexpectationvaluesofthesecorrelationfunctionsaregivenbyEquation(8).
Inordertoincludepolarizationofthetarget,theeffectivetargetfunctioncanbewrittenas:
Φpol(→r ,→r ) = ϕ (→r )+ χST(r1)u1s→p(r2)co√s(θ12), (2)
1 2 0 2 r12 r2 Zπ
whereZisthenuclearcharge. Thefunctionu1s→p isgivenby:
Z
u1s→p(r2) = e−Zr2(2r23+r22) (3)
Thetargetfunctionisgivenby:
(cid:114)
Z3
ϕ (r ) = e−Zr2. (4)
0 2
π
Wefindthatthereisaprobleminusingtheprojectionoperatorswhenpolarizationofthetargetis
includedinthetargetfunction. AccordingtoFeshbach(1962)[2],theprojectionoperatorsshouldbe
idempotent,i.e.,:
P2 = P, Q2 = Q, andPQ =0 (5)
Inordertosatisfytheaboveconditions,theperturbedwavefunctiongiveninEquation(2)must
(cid:42) →
benormalizedtounityforall r and r . Thisisnotpossible,asindicatedbyBransden(1970)[11].
1 2
Therefore, we have not used the projection operators in the present calculation as it becomes too
difficult to write the optical potential and to satisfy the above conditions in the presence of the
polarizationofthetarget.
Another variational approach, also based on perturbation, has been proposed by Drachman
(1968)[12]. Hewritesthewavefunctionintheform:
(cid:42) → → → → →
Ψ(r , r ) = χ (r )ϕ (r )+Φ(r , r ) (6a)
1 2 k 1 0 2 1 2
where:
→ → → → → →
Φ(r , r ) = F(r )G(r , r )ϕ (r ) (6b)
1 2 1 1 2 0 2
→
Thecorrelationfunctioncanbewrittenintheaboveform,whereG(r , r )istheexactadiabatic
1 2
→ →
perturbationincludingallmultipoles. Thefunctionsχ(r )andF(r )aredeterminedvariationally
1 1
→ →
usingG(r , r ),numerically,obtainedfromtheexactanalyticformofG,giveninellipticalcoordinates
1 2
byDalgarnoandLynn(1957)[13].However,retainingtheshort-rangemonopolepartisunjustifiedand
bysuitablysuppressingit,thescatteringlengthandthephaseshiftscomparabletothoseofSchwartz
(1961)[14]havebeenobtained.
ThevariationwithrespecttoC ,asindicatedbyBhatia(2007)[7],inthefunction:
λ
→ → → →
I =< Ψ (r , r )|H−E|Ψ (r , r ) > (7)
L 1 2 L 1 2
Atoms2016,4,27 3of13
Gives:
<V (r )u(r ) >
C = λ 1 1 (8)
λ k2−Z2−ε(λ)
whereV isgivenby:
λ
→ → → →
V (r ) =< Φλ(r , r )|H−E|Φpol(r , r ) >. (9)
λ 1 L 1 2 1 2
andε(λ)istheexpectationvalueofH:
< Φλ|H|Φλ >= ε(λ) (10)
L L
In Equation (2), χ (r ) is the smooth cutoff function introduced by Shertzer and Temkin
ST 1
(2006)[15]. Thistermguaranteesthatχ (r )/r2 →0forr →0. Thisfunctionisgivenby:
ST 1 1 1
1 4
χ (r ) =1−e−2Zr1( (Zr )4+ (Zr )3+2(Zr )2+2Zr +1) (11)
ST 1 1 1 1 1
3 3
Thisfunctionisunlikethestepfunction,ε(r ,r ),introducedbyTemkin(1959)[1],whichequals
1 2
1 when r > r , and equals zero otherwise. This step function ensures that the polarization takes
1 2
placeonlywhentheincidentparticleisoutsidethetarget. Nowwewantthepolarizationfunctionin
Equation(2)tobevalidthroughouttherange,ratherthanonlyforr >r . Theangleθ istheangle
1 2 12
→ (cid:42)
between r and r . ItshouldbepointedoutthatinEquation(2),thereisaplussignafterthetarget
1 2
functionandnotthenegativesign,asinthecaseoftheelectronscattering. Thisisduetothefactthat
theperturbingpotentialisoftheoppositesigntothatinthecaseoftheelectron:
2 2
V(r ,r ) = ±( − ) (12)
1 2
r r
1 12
The positive sign is for positron scattering and the negative sign for electron scattering.
ThecorrectnessofthesigninEquation(2)canbeascertainedfromthepaperofTemkinandLamkin
(1961)[16]. InEquation(1),Listheangularmomentum,u isthescatteringfunction,andthefunction
L
Φ isthecorrelationfunctionwhichcanbewrittenintermsofgeneralized“radial”functions,which
L
dependontheradialcoordinatesandtheEuleranglesintroducedbyBhatiaandTemkin(1964)[17]:
Φ = ∑[fκ,+1(r ,r ,r )Dκ,+1(θ,ϕ,ψ)+ fκ,−1(r ,r ,r )Dκ,−1(θ,ϕ,ψ)]. (13)
L L 1 2 12 L L 2 1 12 L
κ
ForS-wave(i.e.,L=0)D =constantinEquation(13)andΦ istakenofHylleraasform:
L 0
ΦL=0(→r1,→r0) = √1 e−γr1−δr2∑Nω Clmnr1lr2mr1n2 (14)
8π2
lmn
wherethesumincludesalltripletssuchthatl+m+n=ωandω=0,1,2,3,4,and5. Bhatiaetal.
(1971) [9] included a term e−αr12 in the above function to take into account virtual positronium
formation. This term has not been included here, and we have not used the Feshbach projection
operatorseither,inthepresentcalculations. Thescatteringequationwiththeopticalpotentialbecomes
simplerwithouttheuseoftheprojectionoperators. Thewavefunctionofthescatteredpositronis
determinedfrom:
(cid:90) → → →
[Y (Ω )Φpol(r , r )(H−E)Ψ ]dr dΩ =0 (15)
L0 1 1 2 L 2 1
InEquation(15), H istheHamiltonianandEisthetotalenergyofthesystem. Wehaveused
Rydbergunitsthroughout[e2 =2,(cid:125)=1, andm =1/2]. WehaveHintheseunits:
e
2Z 2Z 2
H = −∇2−∇2+ − − , (16)
1 2 r r r
1 2 12
Atoms2016,4,27 4of13
E = k2−Z2, (17)
wherek2isthekineticenergyoftheincidentpositronandZisthenuclearcharge.
ForL=0,wecanwritetheequation(15)forthescatteringfunctionuintheform:
J +J +J +J +J =0. (18)
1 2 3 4 5
where:
J = (cid:90) Y (Ω )ϕ (→r )(H−E)ϕ (→r )u(r1)Y (Ω )d→r dΩ (19a)
1 00 1 0 2 0 2 00 1 2 1.
r
1
1 d2
J = − [ +k2+V (r )]u(r ). (19b)
1 r dr2 d 1 1
1
J2 = (cid:90) Y00(Ω1}ϕ0(→r2)(H−E)χSTr(2r1)u1s→p(→r2)u(rr1)Y00(Ω1)d→r2dΩ1. (20a)
1 1
α(r ) u(r )
J = −χ (r ) 1 1 . (20b)
2 ST 1 (Zr1)4 r1
J3 = (cid:90) Y00(Ω1)χSTr(2r1)u1s→p(→r2)(H−E)ϕ0((cid:42)r2)u(rr1)Y00(Ω1)d→r2dΩ1. = J2. (21)
1 1
J4 = (cid:90) Y00(Ω1)χSTr(14r1)u1s→p(→r2)(H−E)χSTr(12r1)u1s→p(→r2)u(rr11)Y00(Ω1)d→r2dΩ1. (22a)
9 χ (r ) 2 8 χ (r ) 2 u(r ) 43 χ (r )
J = [ ( ST 1 ) − d(r )( ST 1 ) ] 1 + Yp(r ) ST 1 (22b)
4 2 (Zr1)2 3Z 1 r12 r1 8Z6 1 r12
where:
1 χ (r )d2u du 2Z 2 χ (r )
Yp(r ) = − [ ST 1 +2B (r ) +B (r )u(r )+(− − +k2) ST 1 u(r ) (23)
1 r r2 dr2 1 1 dr 2 1 1 r r2 r2 1
1 1 1 1 1 1 1
J = (cid:90) Y (Ω )Φpol((cid:42)r ,→r )|H−E|∑C Φs(→r ,→r )d→r d→r (24a)
5 00 1 1 2 s 0 1 2 1 2
whichcanbewrittenas:
J = ∑ J5S (24b)
5 E−ε(s)
where:
(cid:90) → (cid:42) → (cid:42) → (cid:90)
Js = Y (Ω )Φpol(r , r )|H−E|Φs(r , r )dr dΩ V(r )u(r )dr (24c)
5 00 1 1 2 0 1 2 2 1 s 1 1 1
InEquations(24a)and(24b)thesummationoversisunderstood. SubstitutingEquations(19)–(24)
inEquation(18),wegettheequationforthescatteringfunctionu(r):
[D(r) d2 +k2+V +V −Vpol]u(r) =0 (25)
dr2 d pol opt
Wegivethevariousquantities:
43 χ (r) 2
D(r) =1+ ( ST ) (26)
8Z6 r2
Thedirectpotentialsaregivenby:
2(Z−1) 1
V = − −2e−2Zr(Z+ ) (27)
d r r
Atoms2016,4,27 5of13
and
d
V = (x +x )+x (28)
pol 1 3 2dr
Thevariousquantitiesintheaboveequationare:
43 2Z 2 χ (r) 2
x = (− − +k2)( ST ) , (29)
1 8Z6 r r2 r2
43 χ (r)
x = B (r) ( ST ), (30)
2 1 8Z6 r2
43 χ (r) 2α(r) 9χ (r)2 8 χ (r) 2
x = B (r) ( ST )+ χ (r)− ST + d(r)( ST ) , (31)
3 2 8Z6 r2 (Zr)4 ST 2 (Zr)4 3Z r2
1 2 2 4 2
B (r) = [−2Zχ (r)+2Z− +e−2Zr(− Z4r3− Z3r2+2Z+ ). (32)
1 r2 ST r 3 3 r
4Z 6 4 4Z2 8Z 6
B (r) = −2ZB (r)− + +e−2Zr( Z5r+2Z4− − − ). (33)
2 1 r3 r4 3 r2 r3 r4
Thepolarizabilityofthetargetisobtainedfromthefollowingexpressionforr=infinity:
α(r)/Z4, (34)
where:
9 9
α(r) = −e−2Zr((Zr)4+5(Zr)3+9(Zr)2+9(Zr)+ ). (35)
2 2
d(r)= 129 1 + 18 1 −e−2Zr( 3 r4+ 27r3+ 54 r2+ 135r+ 975 + 1281 1 + 36 1 + 18 1). (36)
32Z5 r Z7r3 16 16 8Z2 8Z3 32Z4 32Z5 r Z6r2 Z7r3
Forr->infinity,x hasaterm:
3
9
, (37)
2(Zr)4
where9/(2Z4)isthedipolepolarizabilityofthetargetwithnuclearchargeZ.
Vpolu(r) =r ∑Nω (cid:68)Y00(Ω1)Φpol(→r1,→r2)|H−E|Φ0s(cid:69)(cid:10)Φ0s|H−E|Ψ0(cid:48)(cid:11). (38)
op 1 E−ε(s)
s
Here,Ψ(cid:48) isthewavefunctiongiveninEquation(1)withoutthecorrelationterm. Thephaseshifts
0
areinferredfromthescatteringfunctionu ≡uforrtendingtoinfinity:
L=0
sin(kr+η)
u(r) = (39)
kr
InTable1,weshowtheconvergenceofphaseshiftsη (radians)fork=0.4withthenumberof
termsinthecorrelationfunctiongiveninEquation(1). ByN=35,thephaseshifthasconvergedto
atleastfoursignificantfigures. InTable2,results,whichhavenotbeenextrapolated,aregivenfor
k=0.1to0.7. Thephaseshiftsobtainedforshorterexpansionsareingoodagreementwiththeprevious
calculationsofBhatiaetal. (1971)[9]. However,theresultsofBhatiaetal. (1971)havebeenobtained
byextrapolationandbyaddingthecontributionofthelong-rangepotentialduetothepolarizationof
thetargettotheextrapolatedresults. Theirfinalresultsdonothaveanyvariationalboundsanymore.
ThepresentandpreviousresultsareshowninFigure1. Wealsocomparethepresentphaseshiftswith
thoseobtainedbySchwartz(1961)[14]obtainedbyusingtheKohnvariationalprinciple,Houstonand
Drachman(1971)[18],byusingtheHarrismethod,KohnvariationalcalculationofHumberstonand
Wallace(1972)[19],andGien(1997)[20],byusingtheHarris-NesbetmethodinTable2. The21-state
Atoms2016,4,27 6of13
close-couplingapproximationphaseshiftsobtainedbyMitroyandRatnavelu(1995)[21–23]arelower
thanthepresentresults. Mitroy(1995)[21–23]hasreportedalargebasisclose-couplingcalculation
ofscatteringatlowenergies. Thephaseshiftsofthesecalculationsarelowerthanthepresentresults
giveninTable2. InthemostrecentpublicationofKadyrovandBray(2016)[24]ontherecentprogress
hasgiventheelasticcross-sectionsatlowpositronenergiesinFigure6. However,extractingphase
shiftsfromthisfigurewillnotbeaccurateenoughforcomparisonwiththepresentcalculations. Phase
shiftsobtainedbyHiggins(1990)[25],usingintermediateR-matrixtheory,arealsogiveninTable2.
Theagreementoftheresultsofthesecalculationsisgood. However,theyarelowerthanthepresent
results. ThephaseshiftsofSchwartz(1961)[14]arewrongattwoincidentpositronmomenta.
Table1. Convergenceofphaseshiftsη (radians)fore+-Hscatteringwithrespecttothenumberof
correlationtermsNfork=0.4.
N γ δ η
10 0.56 0.90 0.09649
20 0.56 0.90 0.11619
35 0.60 0.91 0.11976
56 0.72 0.83 0.12083
Table 2. Phase shifts η( radians) e+-H scattering for various k and comparison with
previouscalculations.
K N Present A B C D E F G
0.1 35 0.14918 0.1483 0.151 0.149 0.148 0.1482 0.142 0.1404
0.2 35 0.18803 0.1877 0.188 0.189 0.187 0.1875 0.182 0.1767
0.3 35 0.16831 0.1677 0.168 0.169 0.167 0.1671 0.159 0.1558
0.4 56 0.12083 0.1201 0.120 0.123 0.119 0.1196 0.111 0.1105
0.5 20 0.06278 0.0624 0.062 0.065 0.062 0.0621 0.055 0.0536
0.6 20 0.00903 0.0039 0.007 0.008 0.003 0.0033 −0.002 −0.004
0.7 20 −0.04253 −0.0512 −0.054 −0.049 ... −0.0520 −0.0588
Atoms 2016, 4, 27  7 of 13 
A:Bhatiaetal.(1971)[9],B:Schwartz(1961)[14],C:HoustonandDrachman(1971)[18],D:Humberstonand
Wallace(1972)[19],E:Gien(1997)[20].F:Higginsetal.(1990)[25];G:MitroyandRatnavelu(1995)[21–23].
0.2
0.15
s) 0.1
n
a
di
a
hift(r 0.05
s
e
s
a
h 0
p
-0.05
-0.1
0.1 0.2 0.3 0.4 0.5 0.6 0.7
incident momentum k
Figure1.(Coloronline)Theuppercurverepresentsthepresentresults,whilethelowercurverepre sents
resultsobtainedbyBhatiaetal.(1971)[9].
Figure 1. (Color online) The upper curve represents the present results, while the lower curve 
represents results obtained by Bhatia et al. (1971) [9]. 
The scattering length is given by: 
1 
limkcot()
(40) 
a
k0
In Table 3 we show the convergence of the scattering length with the number of terms in the 
correlation function given in Equation (1). Temkin (1961) [26] showed that there is a correction due 
to the long‐range correlations: 
1 a a2
a  a(R)(   ).  (41) 
R R2 R3
Table 3. Convergence of the scattering length for e+‐H scattering with respect to the number of 
correlation terms N. 
N     A 
10  0.56  0.90  −1.86341 
20  0.56  0.82  −2.05266 
35  0.56  0.84  −2.10074 
56  0.52  0.84  −2.10158 
The scattering length given in Table 3 for N = 56 has been calculated at R = 1856.470. Only the 
term /R contributes significantly in Equation (41). The final result for the scattering length = 
−2.10158 − 0.002424 = −2.104004 a0. This agrees with the scattering length −2.1036   0.0004 obtained 
by Houston and Drachman (1971) [18], and with −2.103   0.001 by Humberston and Wallace (1972) 
[19], using the Kohn variational principle. Another formula obtained by Drachman (2016) [27] using 
the Calogero method (1967) [28] for a correction due to the long‐range correlation is:
Atoms2016,4,27 7of13
Thescatteringlengthisgivenby:
1
limkcot(η) = − (40)
a
k→0
In Table 3 we show the convergence of the scattering length with the number of terms in the
correlationfunctiongiveninEquation(1). Temkin(1961)[26]showedthatthereisacorrectiondueto
thelong-rangecorrelations:
1 a a2
a = a(R)−α( − + ). (41)
R R2 R3
Table 3. Convergence of the scattering length for e+-H scattering with respect to the number of
correlationtermsN.
N γ δ A
10 0.56 0.90 −1.86341
20 0.56 0.82 −2.05266
35 0.56 0.84 −2.10074
56 0.52 0.84 −2.10158
ThescatteringlengthgiveninTable3forN=56hasbeencalculatedatR=1856.470.Onlytheterm
α/RcontributessignificantlyinEquation(41). Thefinalresultforthescatteringlength=−2.10158−
0.002424=−2.104004a . Thisagreeswiththescatteringlength−2.1036±0.0004obtainedbyHouston
0
andDrachman(1971)[18],andwith−2.103±0.001byHumberstonandWallace(1972)[19],usingthe
Kohnvariationalprinciple. AnotherformulaobtainedbyDrachman(2016)[27]usingtheCalogero
method(1967)[28]foracorrectionduetothelong-rangecorrelationis:
(a(R)−α/R)
a = . (42)
(1−α/R2)
Thisgivea=−2.104007a ,whichisessentiallytheresultobtainedusingEquation(41). Thisresult
0
isconsistentwiththeconvergenceofSchwartz’scalculation(1961)[14]that−2.10>a>−2.11.
Using the presently calculated scattering length, the cross-section at zero energy is given by
σ = 4πa2 =15.57×10−16cm2. IftheexperimentalresultsofZhouetal. (1997)[29]areextrapolated
towardszeroenergy,thentheirresultisfoundtobetoolow. Thepresentresultforthecross-section
shouldbecorrectbecausethescatteringlengthiswellestablishedbynow,havingbeingpreviously
calculatedbySchwartz(1961)[14],HoustonandDrachman(1971)[18],andHumberstonandWallace
(1972) [19]. Since only S-wave phase shifts have been calculated here, a comparison with the
experimentalresultsathigherenergiescannotbemade.
Overtheyears, reviewsonthissubjecthavebeenpublished. AmongthemarebyDrachman
(1971)[30],Bhatia(2014)[31],andKadyrovandBray(2016)[24].
2. Z
eff
In addition to the scattering, there is a possibility of positronium formation and positron
annihilation. The cross-section for annihilation of an incoming positron and an atomic electron
withtheemissionoftwogammaraysisgivenbyFerrell(1956)[32]:
σ (πa2) = Z α3/k (43)
a 0 eff
Atoms2016,4,27 8of13
whereα = e2/(cid:125)cisthefine-structureconstantanda istheBohrradius. AsthequantityZ ,which
0 eff
measurestheoverlapofthetargetelectronwiththepositron,asindicatedbyDiRienziandDrachman
(2003)[33],approachesZ,thenumberofelectronsforafreepositron. Forhydrogen:
→ → → → → →
Z = dr dr |Ψ (r , r )|2δ(r − r ), (44)
eff (cid:120) 1 2 L 1 2 1 2
→ →
ThenormalizationofΨ(r , r )inEquation(1)requiresthatforr → ∞:
1 2 1
1
u (r ) = [4π(2L+1)]1/2k−1sin(kr − Lπ+η) (45)
L 1 1
2
InTable4wepresentvaluesforZ forL=0asafunctionofkbelowthepositroniumformation
eff
threshold. ThepresentresultsarecomparedwiththepreviousresultsbyBhatiaetal. (1974b)[34]and
withthoseobtainedbyHumberstonandWallace(1972)[19]. Theagreementisfairlygood. Greenand
Gribakin(2013)[35],usingthediagrammaticmany-bodytheoryhavecalculatedphaseshiftsandZ ,
eff
givenintheirFigure7. TheirresultsforZ ,giveninTable4,couldnotbereadaccuratelyfromthe
eff
figure. Nevertheless,theirresultsappeartobeclosetothoseobtainedinothercalculations. Itisfound
empiricallythatthevalueofZ atverylowenergiesisproportionaltothedipolepolarizabilityαof
eff
thehydrogenatom(Osmon,1965)[36]andcanberepresentedbyα1.25. Theannihilationofpositrons
withelectronsresultsintwophotonsatarateλ,Fraser(1968)[37]:
λ =4.51×109ρZ /M sec−1 (46)
eff A
Table4.Z (0)asafunctionoftheincidentpositronmomentumk,obtainedinthiscalculationand
eff
comparedwithpreviouscalculations.
k A B C D
0.1 8.092 7.363 7.5 7.5
0.2 5.357 5.538 5.7 5.5
0.3 4.264 4.184 4.3 4.1
0.4 3.370 3.327 3.3 3.5
0.5 2.424 2.730 2.7 3.0
0.6 2.249 2.279 2.3 2.8
0.7 2.069 1.950 ... 2.2
A:presentcalculatons,B:Bhatia(1974b)[34],C:HumberstonandWallace(1972)[19],D:GreenandGribakin
(2013)[35].IntheaboveequationρisthedensityingramspercubiccentimeterandMAistheatomicmass.
3. PositroniumFormation
Positronium, the bound state of an electron and a positron, was predicted by Mohorovicic
(1934)[38]inconnectionwiththespectraofnebulae. Positroniumformationtakesplacewhenthe
incidentpositroncapturesanelectronofthehydrogenatombeforethepositronandelectronannihilate
eachother:
e++H(1s) → Ps+P (47)
wherePsisthepositroniumatomandPistheproton. Experimentaldeterminationofthecross-section
ofpositroniumformationinvariousgaseshavebeencarriedoutbyCharlton(1985)[39]andSteinetal.
(1992)[40]. ExperimentsonhydrogenatomshavebeencarriedoutbyZhouetal. (1997)[29]. R-matrix
calculationshavebeencarriedoutbyChanandFraser(1973)[41],whilecoupled-channelcalculations
havebeencarriedoutbyHumberston(1982)[42]. Usingthedistorted-waveapproximation,Khanand
Ghosh(1983)[43]havecarriedtheircalculationsusingthemethodofpolarizedorbitalsofTemkin
(1959)[1]. ThepresentcalculationsaresimilartothoseofKhanandGhosh. However,thescattering
Atoms2016,4,27 9of13
functionshavebeencalculatedvariationallybyBhatia(2007,2008)[7,8]. Thedifferentialcross-section
forthisrearrangementcollisionfromthegroundstateisgivenby:
∂σ = vPS|T(k ,k )|2 = µi kps|T(k ,k )|2 (48)
∂Ω v i PS µ k i Ps
i f i
Intheaboveequation,v andv arethevelocitiesoftheincidentpositronandofpositroniumin
i PS
thefinalstate,k andk arethemomentaoftheincidentpositronandpositroniumatominthefinal
i Ps
state,andµ andµ aretheinitialandfinalreducedmassesequalto1/2and1,respectively,inRydberg
i f
units. TisthetransitionmatrixwhichisgivenbyKhanandGhosh(1983)[43]andBransden(1965)[44].
T(k ,k ) = −µf < Φ∗ (→r →r )|V |Ψ (→r ,→r ) >, (49)
i Ps 4π PS 1, 2 f L 1 2
where
Φ∗ (→r ,→r ) = η(r )exp[−i→k ·(→r +→r )] (50)
Ps 1 2 12 2 Ps 1 2
Thegroundstatewavefunctionofthepositroniumatomisgivenby:
e−0.5r12
η(r ) = √ (51)
12
8π
Theinteractionpotentialinthefinalstateisgivenby:
Ze2 Ze2 Z Z
V = ( − ) =2( − ) (52)
f r r r r
1 2 1 2
InRydbergunits,beingusedthroughout,e2isequalto2and(cid:125)isequalto1.
Wecanwrite:
T(cid:48)(k ,k ) = gR(k ,k )+gRP(k ,k ) (53)
i PS i PS i PS
The first term above involves the target function only and the second terms involve the
polarizationofthetarget,asindicatedinEquation(2). Theintegrationofthesetermscanbecarriedout
byusingFouriertransforms,giveninKhanandGhosh(1983)[43]andCheshire(1964)[45],ofvarious
termsintheintegrand,asindicatedintheappendix. WeconsiderL=0andthewavefunctionfor
theinitialstateisgiveninEquation(1)withoutthecorrelationterms. Thedifferentialcross-sectionis
givenby:
dσ k 1
(a2) = µ µ PS |T(cid:48)(k ,k )|2, (54)
dΩ 0 i f k 8π i PS
i
whereT’isgivenintheAppendixA.Thetotalcrosssectionisgivenby:
(cid:90) ∂σ 1k
σ(a2) = dΩ = Ps|T(cid:48)(k ,k )|2 (55)
0 ∂Ω 4 k i Ps
i
Inπa2units,thecross-sectionisgivenby:
0
(cid:90) ∂σ 1 k
σ(πa2) = dΩ = Ps|T(cid:48)(k ,k )|2 (56)
0 ∂Ω 4π k i Ps
i
Wehaveusedµ µ =1/2inRydbergunits. Cross-sectionsobtainedusingonlygR(k ,k ),and
i f i PS
usinggR(k ,k )+gRP(k ,k )forpositroniumformationatvariousenergiesaregiveninTable5and
i PS i PS
comparedwiththoseobtainedbyKhanandGhosh(1983)[43]. Thefirstcolumngivescross-sections
without the polarization of the target and their maximum value occurs at k2 = 0.75 while, with
i
polarization, the maximum is at k2 = 0.80. The cross-sections increase considerably when the
i
polarizationofthetargetistakenintoaccount. Itisseenthatthepresentresultsagreewithresultsof
Atoms2016,4,27 10of13
KhanandGhosh(1983)[43]andarealsoinreasonableagreementwiththoseofHumberston(1982)[42].
ThepresentresultsarealsoinreasonableagreementwiththoseofChanandFraser(1973)[41]when
the second term in Equation (53) is not included. This is rather surprising since polarization is
very important in the positron-hydrogen scattering. The present results are also compared with
thoseofKvitsinskyetal. (1995)[46]whousedtheFadeevequationsfortheircalculations. Again,
the presently-calculated cross-sections are only at S-wave, and they cannot be compared with the
experimentalresultsofZhouetal. (1997)[29]. Similarly,higherpartialwavesarerequiredtocalculate
differentialcross-sections.
Table5. Comparisonofthepresentresultsforthepositroniumformationcross-section(π a2)with
0
thoseobtainedinothercalculations.
k2 A B C D E
i
0.5041 0.0010053 0.0066228 0.009037 0.0041 0.0038
0.5476 0.0025753 0.018783 ... ... ...
0.5625 0.0026829 0.020249 0.024795 0.0044 0.0041
0.64 0.0025604 0.022566 0.0248 0.0049 0.0047
0.6724 0.002412 0.022350 ... ... ...
0.7225 0.0021366 0.021456 0.021164 0.0058 ...
0.75 0.0020034 0.020835 0.019707 ... ...
0.81 0.0017211 0.019256 ... ... ...
0.9025 0.0013698 0.016760 ... ... ...
1.00 0.0010916 0.014327 ... ... ...
A:Presentresults,withoutpolarization;B:presentresultwithpolarization;C:KhanandGhosh(1983)[43],
D:Humberston(1982)[42],E:Kvitsinskyetal.(1995)[46].
4. Conclusions
Using the hybrid theory of scattering of Bhatia (2007) [47], we have calculated phase shifts
which have lower bounds to exact phase shifts. This calculation, which includes the contribution
ofthelong-rangeinteraction−1/r4 variationally,requiresfewercorrelationfunctionscomparedto
thepreviouscalculationofBhatiaetal. (1971)[9]. Thescatteringlengthalsoagreeswithprevious
calculationsofSchwartz(1961)[14]andHoustonandDrachman(1971)[18]. Thescatteringfunctions
havebeenfurtherusedtocalculateZ andpositroniumformationcross-sectionsinthedistorted-wave
eff
approximation. Thescatteringfunctionsobtainedusingthehybridtheoryhavebeenusedpreviously
tocalculatephaseshiftsandphotoabsorptioncross-sectionsbyBhatia(2004,2006,and2008)[5,6,8],
givingresultsagreeingwiththoseobtainedusingdifferentmethods. Equation(25)canbeusedto
calculateboundstateenergiesaswell,asindicatedbyBhatiaandMadan(1973)[48]. Itisexpectedthat
thepresentresultsshouldbeaccurate,aswell,uptothefourthdecimalplace.
Acknowledgments:ThanksareextendedtoR.J.Drachmanformanyhelpfuldiscussions,foratableofFourier
transforms, forbringingtomyattentionhispublicationof1968onthistopic, andforcriticalreadingofthe
manuscript.IwouldliketothankHasiRayforpointingoutthepublicationofCheshire(1964)[45]onFourier
transformsduringherstayatGoddardin2015.Calculationswerecarriedoutinquadrupleprecisionusingthe
DiscovercomputeroftheNASACenterforComputationScience.
ConflictsofInterest:Theauthorsdeclarenoconflictofinterest.
AppendixA
Theintegralsoccurringinthecalculationareoftheform:
→
I (r ) = (cid:90) d→r η(r )e−i→kf.→r2ϕ0(r2) (A1)
1 1 2 12
r
2
and
I (r ) = (cid:90) d→r η(r )e−i→kf.→r2/2ϕ (→r ) (A2)
2 1 2 12 0 2