Positronium Formation from Hydrogen
Negative Ion, H
Puspitapallab Chaudhuri
Instituto deFsia,UniversidadedeS~aoPaulo,
CaixaPostal66318, 05315-970S~aoPaulo,Brazil
Reeivedon21Otober,1999. Revisedversionreeivedon5June,2000.
Theoretialresultsforthepositronium(Ps)formationfromhydrogennegativeion,H ,targetsare
presented. AnexatanalytialexpressionisonstrutedforthetransitionmatrixforPsformation
inollisions of positrons with H in the frameworkof Born approximation using threedierent
targetwavefuntions. Theorrespondingdierentialandtotal rosssetionsarepresentedforPs
formationinground state. Comparison between theseross setions for dierent wavefuntions
reveals the dependeneofthe Ps-formation proess onthehoieof thetarget wavefuntionand
theeetoforrelationamongthetargeteletronsontheproess.
I Introdution
Hydrogen atom, being the simplest system in atomi
physis,hasbeenusedextensively in thedevelopment
of quantum mehanis. Despitetherepulsive
Coulom-biinterationbetweeneletronswhihisequalin
mag-nitude to theirattrationto aproton,itis possibleto
attah an eletron to the hydrogen atom. Although
veryweakly bound, the twoeletrons and the proton
togetherformthesinglyhargednegativehydrogenion,
H . Itisthesimplestnegativeion(boundbya
short-rangepotential).
In last few years , there has been a signiant
progressinstudyingollisionproessesinvolvingH . It
isofgreatimportaneinastrophysisasitplaysa
fun-damentalroleinmaintainingtheradiationequilibrium
ofthesolaratmosphere[1℄andalsoprovidesastringent
testoftheapproximationmethodsusedintheanalysis
oftwo-eletronsystems[2℄. Moreover,H alsohas
im-portantbiologial appliation as it an be used as an
eÆientantioxidantin humanbody [3℄. On theother
hand, both positron (e +
) and positronium atom (Ps)
havedierentimportantappliationsinmanydierent
branhesofphysis.Hene,studyingPs-formation
pro-essfrom H bypositronimpatissomethingofgreat
interest.
Reently, Luey et al[4℄studied the(e,2e) proess
on H and showed the sensitiveness of the proess to
the form of the approximationused for the H
wave-funtions. Quiteinthesamespirit,weinvestigatehere
the positronium formation proess by the ollisionof
positronandH (e +
+H !Ps+H)andstudythe
eet of target eletron orrelation on the formation
ross setion. In this ase the nal hannel onsists
onsidered tobein groundstatein the present study.
We onsider three dierentrepresentations of theH
wavefuntion, orrelated and unorrelated versions of
Chandrasekharrepresentation[5℄ and Slater
represen-tation[2,4℄toalulatethedierentialandtotalross
setionforthePs-formationinitsgroundstate. As
dis-ussedintheref. [5℄and [4℄,Chandrasekhar
represen-tationwhih desribesone eletronasstrongly bound
andtheother looselybound to thenuleus,gives
bet-terionizationpotentialandgroundstateenergyforH .
Considerationofthe orrelationbetweentheeletrons
improvestheenergy valueonsiderably. Infat, there
existseveraldierenttypesofH wavefuntion. Very
reently Le Seh [6℄ formulated anew orrelated
ver-sionoftheH wavefuntionwhihgivesslightlybetter
ground stateenergy than that of Chandrasekhar
rep-resentation. However, for the time being, we hoose
to work with Chandrasekhar representation
onsider-ingthegoodenergypreditionandomparatively
sim-ple struture and of the wavefuntion whih redues
theomputational labour signiantly. In table 1We
present the ionization potential and ground state
en-ergy of H due to various wave funtions. The
dif-ferene between orrelatedChandrasekhar
representa-tion and Le Seh representation of H wave funtion
intermsofthepreditionofgroundstateenergyisnot
muhsigniantandhene,wehopetheorrelated
ver-sion of the Chandrasekhar wavefuntion will provide
fairly aurate results. Slater wave funtion for H ,
onthe other hand, predits muh higher value of the
groundstate energy whih atually makes H an
un-stable struture [5℄. Still we inlude this in ourstudy
asit isinterestingto see theeet ofwavefuntionon
Chan-oforrelationinthePs-formationfromH .
Table 1. Ionization potential and ground state energies due to some wavefuntions of H ompared to the
experimentalvalues.
Wavefuntion Ionization GroundState
potential(a.u.) energy(a.u.)
Experimental 0.028 -0.5277
Slater -0.027 -0.473
Chandrasekhar 0.026 -0.5259
(Correlated)
Chandrasekhar 0.014 -0.514
(unorrelated)
Pekeris[7℄ 0.028 -0.5276
Thakkar &Koga[8℄ 0.028 -0.5276
LeSeh 0.027 -0.5266
S.H. Patil[9℄ 0.025 -0.5253
In the present work we use Born approximation
(BA) and evaluate the transition matrix in exat
an-alyti form using Lewis integral [10℄. Although there
exist more rened approximationshemes to estimate
satteringparameters,BAstillhasasigniant
impor-tane in atomi physis. It givesquite auratevalue
whenthetotalorparialrosssetionisnotlarge,as
o-ursgenerallyatthehigherenergies[11℄. Exat
evalua-tionoftheBornsatteringamplitudeisveryimportant
formanyofthesemorerenedapproximationshemes
andusefultohaveageneralviewoftheorresponding
sattering proess. Sine in the present work we are
interested in studying the qualitative behavior of the
eetoforrelationandthedependeneofthe
satter-ing ross setions on the hoie of wavefuntion, the
useBAissuÆient.
Wewill report the dierential and total ross
se-tionsforPs-formationfromhydrogennegativeionin a
energy range of 50-500 eV using both Chandrasekhar
and Slater representation of wavefuntion for H and
ompare them among themselvesto study the
depen-deneoftheproessonthehoieoftargetwave
fun-tionandorrelationamongtargeteletrons.
II Theoretial development
The three dierent wave funtions of our interest are
givenby,
i)Slaterrepresentation:
Sl
H (r
i ;r
j )=
Z 3
e
Z(ri+rj)
(1)
withZ =11=16.
a)Unorrelatedversion:
Unorr
H (r
i ;r
j )=
N
1
4
e
ri rj
+e
ri rj
(2)
withN
1
=0:3948;=1:039and =0:283.
b)Correlatedversion:
Corr
H (r
i ;r
j )=
N
2
4
e
ri rj
+e
ri rj
(1 Dr
ij )
(3)
with N
2
= 0:36598;= 1:0748; = 0:4776 and D =
0:31214:Inthisworkweshallusei=2andj=3.
Inatomiunits,therstBornsatteringamplitude
forPs-formationfromH isgivenby
f B
(k
f ;k
i ) =
1
Z
(r
3 )!
(r
12 )e
ik
f S
12
V(r
1 ;r
2 ;r
3 ) (r
2 ;r
3 )e
ikir1
dr
1 dr
2 dr
3 ;
(4)
where
V(r
1 ;r
2 ;r
3 )=(
1
r
1 1
r
2 +
1
r
23 1
r
13
): (5)
Forthepurposeofstudyingtheinitialtarget-state
ele-tronorrelationonthenalhannelwehoosethepost
formoftheinteration. Inwriting(4)wedenotethe
in-identpositronandtwoeletronsofthetargethydrogen
negativeionas partiles1, 2and 3respetively. Here
(r
2 ;r
3 );(r
3
)arethewavefuntionsofthetargetH
and theneutralhydrogenatom of thenal hannel in
the ground state, respetively and !(r
12
) that of the
positroniumatomwith
S
12 =
1
2 (r
1 +r
2 ); r
12 =r
1 r
2
: (6)
S
12
thepositronium(Ps),k
f
arerelatedthrough
onserva-tionofenergyby
1
2
i k
2
i +E
H =
1
2
f k
2
f +E
Ps +E
H
(7)
where E
H
is the ground state energy of the
hydro-gennegativeion,E
Ps
=-0.25a.u. andE
H
=-0.5a.u.
i
and
f
arethe reduedmasses in the initial andnal
hannels respetively(
i =1,
f
=2). Thus the energy
onservationrelationin thepresentasebeomes,
k 2
i +2E
H =
1
2 k
2
f
1:5 (8)
Toevaluatetheamplitudein (4)weworkwith the
groundstatehydrogenwavefuntion, 1
p
e
r3
and the
ground state positronium wave funtion is taken as
1
p
8 e
r12
j
=0:5
. Also we use the Fourier
transfor-mation
e r
ij
=
2
Z
e
ip(~ri ~rj)
( 2
+p 2
) 2
dp: (9)
an bearranged in terms of Lewisintegral [10℄ whih
hasstandardanalytial solution. The nal expression
ofthesatteringamplitudeforeahwavefuntion
on-sideredisgivenbelow:
i) for Slater representation of the wavefuntion, eqn.
(1)
f B
(k
f ;k
i )=I
1 +I
2
; (10)
where
I
1
= A B; (11)
I
2
= C D: (12)
Nowusingsomegeneralnotations,Q
mn (;
i
;x;y )and
L
lmn (;
i ;
1 ;
f ;
2
) (whose denitions will be given
later)weanwrite
A =
8 p
8Z 3
2
(1+Z) 3
lim
t!0 L
101 (;
i ;t;
f
;Z); (13)
B= 32
p
8Z 3
(1+Z) 3
Q
21 (;
i
;Z ;K); (14)
C= 32
p
8Z 3
(1+Z) 3
Q
21 (;
i
;Z ;K) f(2Z+1)(Z+1)gQ
22 (;
i
;2Z+1;K)
Q
21 (;
i
;2Z+1;K)
; (15)
D= 8
p
8Z 3
2
(1+Z) 3
lim
t!0 L
101 (;
i ;t;
f ;)
1
2
(1+Z)L
111 (;
i
;1+Z;
f ;)
L
101 (;
i
;1+Z;
f ;)
: (16)
ii)forunorrelatedrepresentationofChandrasekharwavefuntion,eqn. (2)
f B
(k
f ;k
i )=I
1 +I
2
(17)
where
I
1
= A+B; (18)
I
2
= C+D: (19)
with
A =
64C
3 lim
t!0 L
101 (;
i ;t;
f ;)
256C 3
3 Q
21 (;
i
B =
64C
(+1) 3
lim
t!0 L
101 (;
i ;t;
f ;)
256C 3
(+1) 3
Q
21 (;
i
;;K); (21)
C=i C
1 i
C
2
; (22)
with,
i C
1 =
256C 3
(+1) 3
Q
21 (;
i
;;K) fY(+1)gQ
22 (;
i
;Y;K) Q
21 (;
i ;Y;K)
; (23)
i C
2 =
64C
(+1) 3
lim
t!0 L
101 (;
i ;t;
f ;)
1
2
(+1)L
111 (;
i
;+1;
f ;)
L
101 (;
i
;+1;
f ;)
; (24)
and
D=i D
1 i
D
2
: (25)
with,
i D
1 =
256C 3
(+1) 3
Q
21 (;
i
;;K) fY(+1)gQ
22 (;
i
;Y;K) Q
21 (;
i ;Y;K)
; (26)
i D
2 =
64C
(+1) 3
lim
t!0 L
101 (;
i ;t;
f ;)
1
2
(+1)L
111 (;
i
;+1;
f ;)
L
101 (;
i
;+1;
f ;)
: (27)
InthisaseY =++1andC=N
1 =8
p
2 3
.
iii)fororrelatedrepresentationofChandrasekharwavefuntion,eqn. (3)
f B
(k
f ;k
i )=I
1 +I
2 I
3
; (28)
where,
I
1
= A+B; (29)
I
2
= C+D; (30)
I
3
= E+F: (31)
A =
64C
(+1) 3
4 2
Q
21 (;
i
;Y;K)+(1+)YQ
22 (;
i ;Y;K)
(1+)
2 L
111 (;
i
;1+;
f
;) L
101 (;
i
;1+;
f
;) (32)
B =
64C
(+1) 3
4 2
Q
21 (;
i
;Y;K)+(1+)YQ
22 (;
i ;Y;K)
(1+)
2 L
111 (;
i
;1+;
f
;) L
101 (;
i
;1+;
f
;) (33)
C=
64CD
(+1) 3
4 2
4Y 2
(1+)Q
23 (;
i
;Y;K)+(1+ 2Y)Q
22 (;
i ;Y;K)
+
(1+)
2 L
112 (;
i
;1+;
f ;)+
(1+)
2 L
111 (;
i
;1+;
f ;)
L
102 (;
i
;1+;
f ;)
1
L
101 (;
i
;1+;
f
(+1) 3
23 i 22 i
+
(1+)
2 L
112 (;
i
;1+;
f ;)+
(1+)
2 L
111 (;
i
;1+;
f ;) L 102 (; i
;1+;
f ;) 1 L 101 (; i
;1+;
f
;) (35)
E=
64CD
(+1) 4
4 2
4Y(1+) 2
Q
23 (;
i
;Y;K)+(5Y 1)(1+) 2
Q
22 (;
i ;Y;K)
+ 3Q
21 (;
i
;Y;K) Y(1+)Q
22 (;
i ;Y;K)
+
(1+) 2 2 L 121 (; i
;1+;
f
;) 1:5(1+)L
111 (;
i
;1+;
f ;) + 3L 101 (; i
;1+;
f
;) (36)
and
F =
64CD
(+1) 4
4 2
4Y(1+) 2
Q
23 (;
i
;Y;K)+(5Y 1)(1+) 2
Q
22 (;
i ;Y;K)
+ 3Q
21 (;
i
;Y;K) Y(1+)Q
22 (;
i ;Y;K)
+
(1+) 2 2 L 121 (; i
;1+;
f
;) 1:5(1+)L
111 (;
i
;1+;
f ;) + 3L 101 (; i
;1+;
f
;) (37)
where Y =++1,C=N
2 =8
p
2 3
andD=0:31214.
The mathematial expression of the sattering amplitude thus beomes more and more omplex from Slater
representation tothe orrelatedversionof Chandrasekharrepresentation. Inwritingalltheexpressions abovewe
useafewgeneralnotationsthataredened below:
f = k f 2 i =k i k f 2
and K=k
i k f ; (38) Q mn (; i
;x;y )=
1 ( 2 + 2 i ) m (x 2 +y 2 ) n (39) and L lmn (; i ; 1 ; f ; 2 )= l l m m 1 n n 2 L 000 (; i ; 1 ; f ; 2 ) (40) with L 000 (; i ; 1 ; f ; 2 ) = Z 1 (p 2 + 2
)(jp+
i j 2 + 2 1 ) 1
(jp+
f j 2 + 2 2 ) dp: (41) L 000
istheLewisintegral[10℄thathasanalytialsolutiongivenby,
L
000 (;q
1 ;a;q
2 ;b) =
Z 1 (p 2 + 2 )(jp q 1 j 2 +a 2 ) 1 (jp q 2 )j 2 +b 2 ) dp = 2 1 2 log (+ 1 2 ) ( 1 2 ) ; (42) = 2 (43) = f(q 1 ~ q 2 ) 2
+(a+b) 2
gq 2
1
+(+a) 2
q 2
2
+(b+) 2 (44) = f(q 1 ~ q 2 ) 2
+(a+b) 2
g+bfq 2
1
+(+a) 2
g+afq 2
2
+(+b) 2
ThegeneralisedLewisintegralsL
lmn (;
f ;
1 ;
i ;
2 )
oforder3(i.e. l+m+m=3)havebeenevaluated
numerially by the proedure of Roy et al [12℄. The
higher order Lewis integrals are evaluated using its
equalitywithDalitzintegral,D
lmn =L
lmn
[12℄where
D
lmn =
2
m+n
m
1
n
2 Z
1
0 dx
1
l
0
1
! 2
+Æ 2
: (46)
with
!=xq
1
+(1 x)q
2
; (47)
2
=x 2
1
+(1 x) 2
2
+x(1 x)jq
1 q
2 j
2
(48)
and
Æ=
0
+ (49)
The Dalitz integrals are arried outby a 16 point
Gauss-Legendre quadrature with subsequent division
of the interval (0,1) to obtain a pre-determined
rela-tive errorof10 10
. Thetotalrosssetions are
alu-lated numerially by using a Gauss-Legendre
quadra-turemethodforintegratingthedierentialrosssetion
overtheosineofthesatteringangle.
III Results and disussion
Based on the exat analyti evaluation of the rst
Born transition matrix we have omputed the
dier-entialross setions(d=d) and total rosssetions
() for the positronium formation using the standard
denitions [13℄ with all the three sattering
ampli-tude mentioned in the previous setion. For brevity,
we denote thedierentialross setions due to Slater
wavefuntionas(d=d)
slater
andthat dueto
orre-latedandunorrelatedChandrasekharwavefuntionas
(d=d)
orr
and (d=d)
unorr
respetively. In Figs
1and2, wepresentaomparativestudy ofthe
dier-entialrosssetionsduetothethreewavefuntions.
In Fig. 1, we plot (d=d)
slater
, (d=d)
orr
and (d=d)
unorr
separately eah for threeenergies,
E=100, 300and 500 eV to havea omparison among
themselveswhileinFig.2,(d=d)
slater
,(d=d)
orr
and(d=d)
unorr
areompared forinidentenergies
E=100,200,300and400eV.Qualitatively,eahofthe
urves with unorrelated wave funtion has the same
featurehavingasharpminimum. As energyinreases,
amplitudeduetotherepulsiveinterationpartaswell
as that due to the attrative part beome more and
morepeakedin the forwarddiretion and theminima
movetowardsthesmallerangles. Theouraneofthe
sharp minimais due to thedestrutive interfereneof
theattrativeandrepulsivepartsoftheinteration
po-tential. However,(d=d)
unorr
shows adierent
na-ture. Unliketheunorrelatedase,theinter-eletroni
repulsionpushesthedipoftheorrelatedwave-funtion
towards higher sattering angles with the inrease of
that the rosssetion dereasesin asmooth
exponen-tial manner. From Fig. 1we an see that in ase of
(d=d)
slater
, the minima are more or less at the
same sattering angle for all energies, the movement
being extremely slow. But in ase of (d=d)
unorr
theminimamovetowardsforwardsatteringanglewith
theinreaseofenergyatafasterrate. Howeverinboth
theasesofunorrelatedwave-funtiontheminimaare
loatedsomewherearound30 Æ
satteringangle.
Figure 1. A omparison of dierential ross setions (in
units ofa 2
0
) for the formationofPs(1s) statefrom H
us-ingSlaterwavefuntion,unorrelatedChandrasekharwave
funtion and orrelated Chandrasekhar wave funtion
re-spetively. Eah blok ontains dierential ross setion
urves for inident positron energy E=100 eV
(dashed-dottedline),E=300eV(dottedline)andE=500eV(Solid
line).
In Fig. 2, we ompare the dierential ross
se-tions for the three wave funtions at four partiular
energies. Here we see, (d=d)
slater
is always a bit
higherthan(d=d)
unorr
forsatteringanglesbelow
20 Æ
and above 30 Æ
at all energies, but the dierene
between them diminishes with the inrease of energy.
However,omparingwith(d=d)
orr
weansaythat
(d=d)
slater
liesmoreorlessloseto(d=d)
unorr
atallenergiesinspiteoftheimportantfatthatSlater
wavefuntion providesapoorervalue ofground state
energyofH thanthatprovidedbyunorrelated
Chan-drasekharwavefuntion (table 1). Inthis ontext,its
worthnotingthat(d=d)
orr
runsalwayslowerthan
(d=d)
slater
and (d=d)
unorr
and the dierene
amongthemissigniant. Heneweanonludethat
to study the Ps-formation from H , onsideration of
im-Figure 2. A omparison of dierential ross setions (in
units ofa 2
0
) forthe formationofPs(1s) statefrom H
us-ing Slater wavefuntion(dottedline),unorrelated
Chan-drasekhar wave funtion(solid line) and orrelated
Chan-drasekhar wave funtion (dashed-dotted line) respetively
atdierentenergies. Inthegure,x-axisdenotesthe
sat-tering angleindegreeswhiley-axis isthe dierentialross
setions.
InFig. 3we displaythe resultsfor thetotal ross
setionforPs-formationingroundstateusingthethree
dierent representations. Unlike the dierential ross
setions, totalross setions due to Slater
representa-tionarealwayshigherthanthatofunorrelated
Chan-drasekharrepresentation,howeverthedierene
dimin-isheswithinreaseofenergy. Thisisduetothehigher
valuesofthe(d=d)
slater
forthelowersattering
an-gles whih ontribute most to the total ross setion
values. Thetotalrosssetionduetoorrelated
Chan-drasekhar wave funtion is muh smaller than both
thetwounorrelatedwavefuntion asexpeted. Sine
there exist no experimental results for apture ross
setions in e +
H ollisions omparison of ourross
setionsisnotpossible.
Inonlusion,wehaveperformedanlosed
analyt-ialalulationofrstBorn transitionmatrixelement
and presentthe dierentialand totalross setion for
positroniumformationingroundstateine +
H
olli-sion fordierent positron impatenergies. A
ompar-isonbetweentheSlater representationandmore
au-rate Chandrasekhar representations, both in
unorre-lated and orrelated version, of wave funtion for
hy-drogen negativeionhasbeendrawnthroughthese
Ps-formationrosssetionvalues. Itislearfromthe
om-parisonthatthePs-formationproessfromH isquite
sensitivetothehoieoftarget wavefuntionand the
onsiderationoftheorrelationeetbetweenthe
ele-tronsoftargetH isveryimportant. Itsbeyonddoubt
thattogetmorelearpitureofthepartiularproess,
more rened models and approximations are required
whihwewillonsiderinourforthomingstudies.
How-ever,thepresentmethod of analytialalulationwill
advanedstudies. Moreover,webelievethatthepresent
endeavourwillhelptoprovideageneralideaaboutthe
Ps-formationproessfrom hydrogen negativeion, H
and a lear visualization of the eet of the eletron
orrelationonthisrearrangementproess.
Figure 3. Total ross setions (in units of a 2
0
) for the
Ps(1s)-formationfromH usingSlaterwavefuntion
(dot-tedline),unorrelatedChandrasekharwavefuntion(solid
line)andorrelatedChandrasekharwavefuntion
(dashed-dottedline)respetivelyatdierentenergies.
TheauthorwouldliketothankProfA.S.Ghoshfor
useful disussions and FAPESP for the nanial
sup-port.
Referenes
[1℄ S.Chandrasekharand D.D. Elbert,Astrophys. J128,
633(1958).
[2℄ B.H. Bransdenand C. J.Joahain, Physis of Atoms
andMoleules(London: Longman,1983).
[3℄ www.nutribiz.om/h ions.html
[4℄ S.Luey,C.T.Whelan, R.J.AllanandH.R.J.
Wal-ters,J.Phys.B:At.Mol. Phys29,L489(1996).
[5℄ S.Chandrasekhar,Astrophys.J100,176(1945).
[6℄ C.L.Seh,J.Phys.B:At.Mol.Phys29,L47(1997).
[7℄ C. L. Pekeris, Phys Rev 115, 1216 (1959), 126 1470
(1962).
[8℄ A.J.ThakkarandT.Koga,PhysRev.A50,854(1994).
[9℄ S.H.Patil,Euro.Phys.J.D6,171(1999).
[10℄ R.R.Lewis,Jr,PhysRev102,537(1956).
[11℄ J.W.Darewyh,J.Phys.B:At.Mol.Phys.20,5925
(1987).
[12℄ U.Roy,U.J.Dube,P.MandalandN.C.Sil,Comput.
Phys.Commun.92,277(1995).
[13℄ BransdenBHandJoahainCJ,Introdutionto
