Cross section of high energy proton alka

I L NUOVO CIMENTO

Vor,. 54 B, N. 1

11 Novembre 1979

Cross-Section of High-Energy Proton-Alkali Atom Charge
Exchange in the Eikonal Approximation (*)(**).
X~. ])ANIELE, G. FERRANTE and E. FIORDIL~0
Istituto di ~Visica dell'Universit& - Via Archira]i 36, 90123 Palermo, Italia
(ricevuto il 16 Marzo 1979)

Summary.
-The eikonal-approximation treatment, developed previously within the framework of the time-dependent impact parameter
method for high-energy proton-alkali atom charge transfer, is applied
to calculate t o t a l cross-sections of valence electron capture into the
hydrogen atom ground state from the whole series of the alkali atoms.
The l a t t e r are treated as one-electron model atoms b y means of very
simple model potentials and wave functions. The calculated crosssections arc compared with the corresponding quantities calculated in
the OBK and first Born approximations. The eikonal results are found
to be lower t h a n 0 B K and F B A results. The similar t o t a l cross-section

of charge transfer in proton-hydrogen collisions is also calculated as a
test of the theory. The resulting hydrogen cross-section is slightly above
the measured cross-section. F o r the hydrogen case a disagreement is
found with the calculation b y Dewangan within the same eikonal
treatment.

1.

-

Introduction.

I n a p r e v i o u s p a p e r (1)7 h e n c e f o r t h t o b e r e f e r r e d t o as p a p e r I , w e h a v e
g i v e n a n e i k o n a l - a p p r o x i m a t i o n t r e a t m e n t w i t h i n t h e f r a m e w o r k of t h e t i m e d e p e n d e n t i m p a c t p a r a m e t e r m e t h o d t o c a l c u l a t e t o t a l c r o s s - s e c t i o n s of h i g h -

(*) To speed up publication, the authors of this p a p e r have agreed to not receive the
proofs for correction.
(**) P a r t i a l l y supported b y GNSM-CNR and CRRNSM.
(1) G. FERRANTE and E. FIORDILINO: High-energy proton-alkali atom charge exchange
i n the eikonal approximation (to be published).

185

186

~. DAN1ELE, G. FERRANTE and ~. FIORDILINO

ener~'y proton-alkali a t o m collisions. The ~iven t r e a t m e n t is an extension to
collisions with alkali atoms of a similar t r e a t m e n t firstly given b y DEWANGAN (2)
for p r o t o n - h y d r o g e n a t o m charge transfer. I n paper I the general t r e a t m e n t ,
a specification of it to the proton-lithium a t o m collisions and the m o t i v a t i o n
to study proton-alkali a t o m collisions were given. I t is the aim of this paper
to present dett~iled calculations based on the above t r e a t m e n t for proton-alkali
a t o m collisions transferring the alkali a t o m valence electron to form groundstate hydrogen atoms. The particular problem at hand, involving only valence
electrons, allows us to circumvent the many-electron character of the alkali
atoms by introducing simple model potentials and wave functions, A completely different situation is encountered when the i m p o r t a n t case of collisions
transferrin~ core electrons is to be considered. However, this case is out of the
scope of this pa,per. An additional aim of this work is to calculate the total
cross-section of p r o t o n - h y d r o g e n collisions transferring the electron to form
ground-state hydrogen atoms. This calculation is m e a n t essentially to provide
a test of the theoretical t r e a t m e n t and a cheek to the entire se~ of calculations.

I n sect. 2 we obtain in closed form the expression for the total capture crosssections. Section 3 is devoted to c o m m e n t the results of the calculations and
to a few concluding remarks.

2.

-

Basic

formulae.

To ealcul~te the total cross-section of proton-alkali a t o m collisions transferrino" the alkali atom valence electron to form gromld-state h y d r o g e n atoms,
we need to specify the valence electron ground-state wave function and the
model potential the valence electron is acted upon in the alkali atom. Here
we choose the model wave function to be hydrogenic:
(1)

~b~(r4) = ~" exp [--yAr~],

and the model potential to be local, H e l l m a n n - t y p e :

(2)

V~(rA) = - L [1 rA

c exp [-

~,',,]].

The combination of eqs. (1) and (2) to simulate the alkali a t o m v~lenco
electron behaviour has been used in the past b y several authors (3,~), and several
(5) ]). P. I)EWANGAN: J. Phys. B, 10, 1083 (1977).
(3) a) G. A. IIART artd P. L. GOODFRIEND:J. Chem. Phys., 53, 448 (1970); b) L. SzAsz
and G. Mc GL~.x: J. Chem. Phys., 42, 2363 (1965); c) H. HELLI~IANN:Acts Physicochim,
USSR, 1, 913 (1935).
(4) a) P. C.~V~LIEP~]~, G. FE~RANTE and B. M. MONTES: Chem. Phys. Lett., 36, 583 (1975) ;
b) B. M. 5IONTES, P. CAVXLIEREand G. FERRX~-TE: Chem. Phys. Lett., 32, 469 (1975);
c) G. FERRANTE, C. LEONE an4 L. Lo CASCIO: Nuovo Cimento B, 51, 91 (1979).

CROSS-SECTIO)I OF YIIGtt-]~NERGY P R O T O N - A L K A L I ATOM CHAI{GE ]EXCHANGE ETC.

187

sets of different parameters ~ , C and a are available (a). Below we will use
the parameters given by t t ~
and GOODFI~IES~D(a). The parameters for lithium,
not given in ref. (~), are instead taken from Schwarz work (6). In this case
the model wave function is

(~a)

~b,(r~) : ~Vir~ exp [-- 7~r~],

as given in paper I.
The combination of eqs. (1) and (2), though very simple and of limited
accuracy in atomic-structure calculations, has proved to give a viable picture
of the alkali atom in collision problems (4).
I n eqs. (1) and (2) and below the target positive ion and the incoming proton are denoted by A and B, respectively. Accordingly, r~ and r. stand for
the distances of the valence electron from the target positive ion and from the
incoming proton. Further, the subscripts i and f will denote ~he atomic initial
and final states. As we are interested in formation of ground-state hydrogen

atoms, the final-state bound-state wave function is

(3)

(/),(rB) = IVf exp [-- 7~rB].

~inally, ~he total cross-section of valence electron transfer from ground-state
alkali atoms to ground-state hydrogen atoms in proton-alkali a t o m collisions
is obtained in the form (see eq. (22) of paper I)

(4)

g,f -~ 4ze2f ]G~f(q, ~ ) -

CGif(q, r~ + a)[2d~q

with

(5a)
(5b)

and

a=7~+\v

2 /'

(5c)
.(W

1)

(5) G.A. H~_~T and P. L. GOODFRIEND: J. Chem. Phys., 53, 448 (1970).
(~) W. H. E. ScHw~z: Aeta _Phys. Aead. Sei. Hung., 27, 391 (1969).

188

1l.

DANIELE,

G.

,~tld

FERRAN'I'E

E.

FIORDILINO

22+@

2 - -- --F(I

4- i~)N~N~v~'J -~ ,

(5d)
92 3 + " 1

1"(1 4- b])N, Ni (1 4- i~])r",,

W = e x - - e , , ~ ] - 1/r. e, a n d eB are, respectively, the g r o u n d - s t a t e energies
of t h e alkali a n d of t h e h y d r o g e n a t o m s ; r is t h e i n c o m i n g - p r o t o n velocity.
To intek~ra.te eq. (4), we firstly c a r r y o u t t h e i n t e g r a t i o n of the t w o t e r m s
n o t c o n t a i n i n g the cross p r o d u c t with the result

(6)

f

(

9a-"-:vi-~l
" 2 , 1 : 6 ; 1 -9 '

(I

(2~,)* + 2*(,>)'a-'-iv. ~F 2 , 1 ; 5 ; 1 - -

+

a,lld

(:)

- 2 IT -21.' 2 , ] ; 4 " 1 ,
IGif(q, 7.4 4- ~)i'-' d~q = ~ ~1 - .[z~ -i-a

5-~'T]-~l:
9
2,1;6;1--

5 /4-4

--

4-~

(2('~.4-2.('~)(~-~Ivl-4]; 2 , 3 : 5 ; 1 - -

,

Y' being t h e h y p e r g e o m e t r i c function.
To p e r f o r m the i n t e g r a t i o n of the cross t e r m

(s)

x:,(q, ~,~, ~,~ 4- ~) = [G *A q , v.,) G,(q, 7.4 + c~) 4- ~1.(q, 7.~) Gj*(q, yA 4- ~)],

we introduc.e t h e h i e r a r c h y
Jo = - -

[1/(a

-- a)] ill ( a / 5 ) ,

J~ - - (112A)(1 -- B J o ) - - In (6'-ta'),

(9)

J . = -- (1/.4)[--(1/6) + BJ~ 4- J o ] ,
J3 - - -- ( l / A ) [ - - (1/25 ~) + BJ2 4- j d ,
J4 = -- ( l / A ) [ - - (1/363) _L B j 3 _~_ J2]

with

a' = a a ,

(9a)

a - aa-

6-

]Tj ~,

ITI~(~ + a) + tTI~ ,

B=a4-O--2IT[

~.

W i t h the help of eqs. (9), integTating the cross p r o d u c t (8) one finds
(10)

f x A q , yA, 7,, + ~)d2q

-- 2~[}212J2 @ ]0)]2J4-} - ()~o~* + ),*(o)J3] .

CROSS-SECTION

OF ttIGH-:ENERGY

PROTON-ALKALI

A T O M CttAI~G:E : E X C H A N G E :ETC.

189

Summarizing the results given b y eqs. (6), (7) and (10), the t o t a l crosssection is obtained in the following closed form:

(~i~: 47~3(~ [).]21v[-2[a-2F (2,1; 4;1--1v-v-~-:)§ C25-~.

(11)

F9

2,1;4;1-

+g

1~[2[~[-0 a - 2 F 2 , 1 ; 6 ; 1 -

(

§

2,1;6;1--

§247

+

[ (

a-~F 2 , 1 ; 5 ; 1 - -

§

§ C2~-2F (2,1; 5;1-- ]v~:)J-- 2C[,,~,2J~ § ,a)12J, § (~a~*§ ~*~o)J.]}.
The corresponding total cross-section for proton-hydrogen a t o m collisions
follows directly from eq. (4) b y letting C ~ 0, N~ ~ Nf ~ 1/V~, y~ ~-- yB ~ 1
(in atomic units), and results to be

§

§247

2,1;6;1--

2

W e observe t h a t eq. (12) differs from the result derived b y DEWANGA~
within the same theoretical t r e a t m e n t (2). We believe the discrepancy to be
due to an oversight b y t h a t a u t h o r in the course of the derivation of the t o t a l
cross-section. However, as is shown b y the plotted cross-sections (fig. 4), our
and Dewangan's cross-sections differ significantly.
H we consider t h a t ~ and to entering the cross-section (12) are functions of
-~ 1/v (see eqs. (5C)), the O B K limit is obtained from eq. (12) b y letting ~ --> 0:
(13)

~
~7--~0

-

~2 a
1
5v 2 (1 § ~ v2) 5

T h a t ~ --> 0 is the limit to be t a k e n to obtain the O B K approximation is seen
b y the expression of the scattering amplitude of the process (eq. (10) of paper I).

3. -

Results

and

comments.

We have calculated the total cross-sections for electron capture into the
ground-state hydrogen a t o m in collisions of protons with alkali atoms in their
g r o u n d state for an energy of the relative motion ranging from 10 to 200 keV.

190

R. DANIEL.E, G. FERRANTE and E. FIORDILINO

TABLE I . - p + L i - - + L i + + H ( l s ) total cross-sections,
proton energy, i n keV, i n diMerent a p p r o x i m a t i o n s .
E

a, i,t~ za2o units,

rs. the relative

Li
~eik

O*OBK

(TFBA

70

0.11975 (--1)

0.78162 (--1}

0.51665 ( - - 1 )

80

0.86291 (--2)

0.26978 (--1)

0.17832

--1)

90

0.64477 ( - - 2 )

0.11458 (--1)

0.75737

--2)

100

0.48947 (--2)

0.68400 (--2)

0.45212

--2)

110

0.377 48 (-- 2)

0.54443 (-- 2)

0.359 86

- - 2)

120

0.29358 ( - - 2 )

0.48988 (--2)

0.32381

--2)

140

0.18760 ( - - 2 )

0.41165 (--2)

0.27210

--2)

160

0.12191 (--2)

0.32779 (--2)

0.216(17

--2)

180

0.83367 ( - - 3 )

0.25032 (--2)

0.16546

--2)

200

0.56149 (--3)

0.18730 (--2)

0.12380

--2)

(*)

D a t a f r o m ref. (D.

10
~t

\
10-1

\

~
o

\

'~ io~2 _-

\

1N#|

.v 0

I

I

I

L

I

50

L

'~

\

I

I

I " I

I

t

I

I

I

100
150
energy (keV)

I

I

i

I

I

I

200

I~'ig. 1. - p ~ I , i - ~ l , i ~ ~ I / ( l s )
total (.ros.~-section, ~, ill ~ao2 units, rs. th(' i'(,i~tiw~ pro~ou
energy, in keV, in different approximations. Continuous curve this work, dashed curve
~)BK result,% ref. (v).

(7) a . FERRANTE, E. FIORD]LLN'O and M. ZARCONE: Vale~ce and core electron capture
il, proto~-alkali atom collision.s (to be published).

CROSS-SECTION OF HIGH-ENI~RGY PROTON-ALKALI ATOM CHARGE EXCHANGE ETC.
TABLE I I . - p ~ - N a - + N a + ~ H ( l s ) a n d p ~ - K - ~ K + - ~ H ( l s ) total cross-sections.
tions as in table I.
E

191
Defini-

Na

~eik

~BK

(~BA

70

0.13504 (--1)

0.14122 ( 0 )

0.93346 ( - - 1 )

8O

0.85855 (--2)

O.58815 (-- 1)

0.38876 (-- 1)

90

0.57334 (--2)

0.25952 (--1)

0.17154 (--1)

100

0.39834 (--2)

0.11749 (--1)

0.77660 ( - - 2 )

110

0.28848 (--2)

0.54620 (--2)

O.361O4 ( - - 2 )

120

0.21715 (--2)

0.26091 (--2)

0.17246 ( - - 2 )

130

0.16715 (--2)

0.12952 (--2)

0.85612 ( - - 3 )

140

0.13151 (--2)

0.68678 (--3)

0.45396 (--3)

160

0.85541 (--3)

0.27743 (--3)

0.18338 ( - - 3 )

200

0.42107 (--3)

0.16803 (--3)

0.11106 ( - - 3 )

E

K

70

0.33418 (--2)

0.64472 (--1)

0.42616 ( - - 1 )

80

0.21390 (--2)

0.23499 ( - - 1)

0.15532 (-- 1)

90

0.14809 (--2)

0.86771 (--2)

0.57355 (--2)

100

0.10862 (--2)

0.34992 (--2)

0.23130 (--2)

110

0.81830 (--3)

0.14238 (--2)

0.94113 ( - - 3 )

120

0.63421 (--3)

0.62224 (--3)

0.41130 ( - - 3 )

130

0.50283 (--3)

0.31496 (--3)

0.20819 (--3)

140

0.40536 (--3)

0.19962 (--3)

0.13950 ( - - 3 )

190

0.15908 (--3)

0.11567 (--3)

0.76457 ( - - 4 )

20o
(*) Data from ref. (s).

(8) M. ZARCONE: private communication.

192

1~. DANIELE, G. FERRANTE and E. I~IOtiDILINO

TABLE I I I . - p + Rb -+ Rb + + II(ls) a~d p =- Cs -* Cs + + H(ls) total cross-sections. Definitions as in table I.
1~"

Rb
Gelk

GOBK

GFBX

80

0.10628 (--2)

0.22591 (--1)

0.14932 (--1)

90

(t.66548 (--3)

0.881 72 (--2)

0.58281 (--2)

100

0.44381 (--3)

0.36111 (--2)

0.23869 (--2)

110

0.31536 (--3)

0.15073 (--2)

0.99632 (--3)

120

0.23180 (--3)

0.64387 (--3)

0.42559 (--3)

130

0.17825 (--3)

0.23834 (--3)

0.15754 (--3)

140

0.14148 (--3)

0.13417 (--3)

0.88686 (--4)

150

0.11426 (--3)

0.72283 (--4)

0.47779 (--4)

170

0.77658 (--4)

0.37488 (--4)

0.24779 (--4)

200

0.46869 (--4)

0.29808 (--4)

0.19703 (--4)

J2

Cs
0"*
OBK

~r
~FB~k

O'eik

~0

o.58227

--3)

o.16421 (-- i)

o.1o854 (-- 1)

90

o.35575

--3)

o.62474 (--2)

o.41295 (--2)

100

0.23736

--3)

0.24726 (--2)

0.16343 (--2)

110

0.16726

--3)

0.10064 (--2)

0.66523 (--3)

120

0.12397

--3)

0.41923 (--3)

0.27711 (--3)

130

0.96071

--4)

0.18010 (--3)

0.11904 (--3)

140

0.72606

--4)

0.57907 (--4)

0.38276 (--4)

160

0.49003

--4)

0.25769 (--4)

0.18223 (--4)

200

0.24738 (--4)

0.17172 (--4)

0.11351 (--4)

(*) Data from ref. (~).

OROSS-SECTIOlq

OF HIGH-ENERGY

PROTON-ALKALI

A T O M CHARG]~ ] E X C H A N G E ]~TC.

193

The transfer of valence electron only is considered, l~or )Ta, K, Rb and Cs
the parameters of the model potentials and wave functions are taken from
ref. (5), while for Li t h e y are taken from ref. (~). The results of the calculations
are r e p 0 r t e d i n tables I-III and in fig. 1-3, together with the corresponding
cross-sections calculated in the O B K and first Born approximation (~). The

10 0

\

L

I
t

10 -1

t

\
\
I

t

\

'\

-9
-~ 10-2
I

10-3

~\

b)
u

10-4

I

I

I

I

I

50

i

i

i

i

I

100

$

i

i

i

i

150

i

i

i

i

i

200

i

50

100

150

200

energy (keY)

Fig. 2. - a) p+Na--~Na++H(ls) and b) p + K - - ~ K + + H ( l s ) total cross-sections, a,
in za~ units, v s . the relative proton energy, in keV, in different approximations.
Continuous curve this work, dashed curve OBK results, ref. (8).

I~BA results are obtained b y scaling the O B K ones b y 0.66 as indicated b y
M_~L~,~O~ (10) (see also ref. (~)).)To attempt is made to compare the calculated
cross-sections with the measured ones, as the latter are total cross-sections
including both captures into excited states and captures from core states (T,11).
Experimental cross-sections are instead compared with our calculations in the

(9) )/L R. C. M:C DOWELL and J. P. COLEMAN: Introduction to the Theory o/Ion-Atom
Collisions (Amsterdam and London, 1970).
(zo) R. A. M~'L:ETO~: Phys. Rev., 122, 1477 (1962).
(it) A. V. VISOGR~OV and V. P. SHV,V~L'KO: Soy. Phys. JETP, 32, 323 (1971).
1 3 - II Nuovo Cimento n .

l~

R. DA.NIELE,

O. F E R R A N T E

~n(~ E. FIORDILI~O

h y d r o g e n a t o m case, where the comparison is possible (fig. 4). F o r this cuse,
D e w ~ n g a n ' s results are also plotted. Figure 4 shows t h a t our eikonal results c o m p a r e well with the m e a s u r e d cross-sections. F r o m the r e p o r t e d e~lculations, it is found the eikonal results to be considerably below the F B A
ones u p to a n energy ~bout 100 keV. F o r greater energies (except in t h e lithium
a t o m ease) the eikonal cross-sections result to be larger t h a n the F B A crosssections. The different b e h a v i o u r of the lithium a t o m cross-sections m a y b e

10"-1

\

10-z
i

d
v
c
.o _
'~10 3

I

\

I

10- 4

\x

b)

10-5

I

[

I

r

I

50

l

I

I

[

J

100

I

I

[

f

I

J

150

I

f

J

I

I

200

0

50

100

150

200

energy ( k eV)

Fig. 3 . - a) p ~ - R b - - + R b + - F H ( l s )
Definitions as i~t fig. 2.

,~nd

b) p ~ - C s - - ~ C s + + H ( l s )

t o t a l cross-sections.

due to our using different model p o t e n t i a l and w a v e function. I f it is true, it
m e a n s t h a t the cross-sections are quite sensitive to t h e model picture of t h e
alkali a t o m a n d t h a t charge transfer collisions m a y provide valuable inform a t i o n to help in determining physically effective model potentials a n d w a v e
functions. A p a r t f r o m this point, the calculated cross-sections are to be considered more accurate t h a n the corresponding F B A cross-sections, as the l a t t e r
are k n o w n to give overestimates a t relatively low energies, and to decrease
too a b r u p t l y with increasing energy.
W e believe the reported results to confirm t h a t the eikonal t r e a t m e n t given

CROSS-SECTION OF H I G H - E N E R G Y PROTON-ALK/LLI ATOM CHARGE E X C H A N G E ~TC.

195

102

10 1

@
v

c:
.o

:~
10 0
(b
o

%%

9

10-I

10-2

I

0

I

i

i

i

50

l

l

i

i

i

i

100

i

a

l

I

i

' 150

I

i

i

i

I

200

energy (keY)

Fig. 4. - p + H ( l s ) - * p % H ( l s ) total cross-sections, s, in r~ao
~ units, vs. the relative
proton energy, in keV, in different approximations. Continuous curve this work, dashed
curve Dewangan's results, ref. (2); 9experiment, ref. (1~).

in p a p e r I m a y prove an effective tool to calculate a v a r i e t y of capture crosssections involving different states in proton-alkali a t o m collisions. However,
more extensive and differentiated calculations are required.

The authors express their t h a n k s to the University of Palermo C o m p u t a t i o n
Centre for the c o m p u t e r time generously provided to them. Stimulating discussions with Dr. M. ZARCONE are also acknowledged.

(15) H. TAWARAand A. RUSSEK: l~ev. Mod. Phys., 45, 178 (1973).

9 RIASSUNTO
L'approssimazione eikonale, sviluppata nell'ambito del metodo del parametro d'impatto dipendente dal tempo, 6 applieata al proeesso di scambio di cariea protone-atomo
alealino. Si caleolano le sezioni d'urto totali di cattura dell'elettrone di valenza dallo

196

R. DANI]~LE, G. F]~RRANT:E fllld ]~. FIOI~DILINO

s t a t o f o ~ d a m e n t a l e d e l l ' a t o m o alcalin