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1/4

V O L U M E

35, NUMBER 17

PH Y SICA L

REVIEW LETTERS

27 OCTOBER 1975

publ ished) .

13

This

number and its associated error are taken to

span

all modern experimental results with which the

a u th o r s are familiar

14

Y

0

K Kim and M. Inokuti, Phys

0

Rev

0

j ^65 , 39 (1968).

15

A.J

0

Rabheru, M. M

0

Islam, and M

0

R

0

Co McDowell,

to

be published

0

16

A

0

L

0

Sinfailam and R. K Nesbet, Phys. Rev. A_6,

2118

(1972).

Various elaborate methods have been employed

to obtain accurate positions and decay widths of

resonances in electron-hydrogen-atom collisions.

1

The most careful and thorough study of this sys

tem is the close-coupling calculations by Burke

and co-workers.

2

Besides the numerous Fe sh-

bach resonances below the n-2 threshold of H,

they predicted the existence of a narrow shape

resonance of

X

P symmetry, with energy only 18

meV above the n = 2threshold. This shape re so

nance greatly influences the Is

2s and Is -

2p

excitation cross sections in electron-hydrogen

scattering

2

and also the photodetachment cr os s

section of H nea r the

n = 2

threshold.

3

As is well known, a shape resonance can be

produced from potential scattering if the poten

tial possesses a barrier at large distances

R

from the force center and is attractive enough at

smallR. In electron-hydr ogen scat terin g, the ex

istence of potential barriers at large

R

in certain

channels has been known for many years.

4

Be

cause of the complicated interactions at small R,

however, it has not been possible to get an est i

mate of the strengths of the potential wells, if

any, at small R, so as to connect the two regi ons .

In this note, we will show that, by using hyper-

spherical coordinates, it is possible to approxi

mate the electron-hydrogen scattering by one-di-

17

F

Q

Jo de Heer and R H. J. Jansen, Fundamenteel

Onderzoek

der Materie Report No.37173 , 1975 (un

published).

18

D .

Andrick and A. Bitsch, J. Phys

0

B: At. Mol.

P h y s .8, 393 (1975).

19

The

derivation given in Refs. 3 and 4 takes this

point

of view in that it essentially obtains a dispersion

r e l a t i o nfor /( E) =/ (E )- \f

Bi

d

E)

- ^ ( E ) ] ,

using prop

e r t i e s of the full Green's function.,

mensional (ID) potential scatt ering . In par ticu lar,

for

l

P states, three potential curves associated

1

with the n = 2 state of H ar e obtained, one being

completely repulsive at all

R

, one having the

property that it can produce a shape resonance,

and one having the ability to support an infinite

number of Feshbach resonances.

In a recent paper

5

(to be called I), I have used

hyperspherical coordinates to study the proper

tie s of doubly excited sta tes of helium. In this co

ordinate system, the distances of the two elec

trons from the nucleus

r

x

and r

2

are replaced by

a hyperradius R = r

x

2

+r

2

2

)

1/ 2

and a hyperangle a

=

arctan(r

2

/r

1

). The angle a, together with the

usual polar coordinates

d

19

cp^)

and

6

2

,cp

2

)

of the

two electrons r epresented collectively as 2= { ,

0

19

cp,0

2

, p

2

}, identify the orienta tion of the elec

tron pair on a 5D spherical sur face, whereas the

coordinateR measures the size of the system.

An interchange of the positions of elec trons 1 and

2 amounts to changing

a

into

\n

-

a

and inter

changing (flu^x) and 0

2

,

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VOLUME 35, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 27 OCTOBER1975

t wo -e l ec t ro n s ys t em t o

[ d V d R

2

- F

p

( R ) + W ^ ( R ) + ^ ( 2 )

Equat ion (2) has a s t ru ctu re s im i lar to that of molecu lar -w ave-fun ct ion expans ion for d ia tomic mo l

e c u l e s , w h e r e V^iR) is the poten t ial cur ve and W^^ifJR) i s the coup l ing te rm between channel s fia nd \i

f

.

If the coupl ing term W^^t can be neglected in the f i rs t app rox ima t ion , as i s indeed the case in the c al

cu la t ion of he l ium potent i a l curves

6

and in the present calculat ion, Eq. (2) reduces to a family of ordi

nary d i f feren t i a l equat ions s im i lar to the s ing le- par t i c l e rad ia l Schrbd inger equat ion wi th poten t i a l

U

ll

R)

=V

ll

R)

- Wj^CR). The e lect ron -hyd rogen sc at te r ing i s thereby redu ced to poten t i a l sca t t e r in g

with potent ial U fo r each channel / i, and the p ro per t i es of bound s ta te s or sc at te r in g s ta te s for each

channel

\i

ar e r e la te d d i re ct ly to the shape of the poten t i a l

U^fjl)

and the behavior of the channel func

t ion $

M

(R; ft).

In con t ras t to the c lose-coupl ing form ulat ion , the poten t i a l U^R) i s local . The sym me try condit ion

on the two-e lect ron wave funct ion

ip(Jl,Sl)

imposed by the Pau l i exclus ion p r incip le i s exp l i c i t ly includ

ed in the channel function ^ ( R j t t ) and the poten t i a l curve U^R). In the actual calcu la t ion , for s ta tes

wi th to ta l o rb i ta l angu lar momentum L and total spin S, I expanded

S

gl

1

l

2{

l iR ^-0l

C

\i l

2

l

1

LM

i

ri^2 \

3)

w h e r e l

1

and l

2

a re t h e an g u l a r -m o m en t u m q u an

tum numbers of the two e lect rons and y

t t L M

r

1 9

r

2

) i s the to ta l o rb i ta l angu la r -m ome ntum wave

funct ion. The sum ma tion over [Z ^ J in (3) is not

ord ere d ; i . e . , [ ^ Z j i s no t d i s t ingu i shed f rom

[Z

2

Zj. In the calc ulat ion for

X

P s t a t e s , t h e s u m

mat ion was t runcated to include only [l

x

l

2

] =[1 ,0 ]

and [ 2 , 1 ] , cons i s ten t wi th the t e rms cons idered

i n t h e t h re e - s t a t e c l o s e -co u p l i n g ca l cu l a t i o n s .

Wi th th i s expans ion , the am pl i tude s g

l l l 2 } 1

(R; a )

at each

R

sat isfy a system of coupled differen

t ia l equat io ns , in which the num ber of equat ions

equal s the num ber of t e rm s re ta ined in the ex

pansion in (3). The eigen value s F

M

(# ) and ampl i

tudesg

tl

i

2ll

(ft;a) a re solved mo st convenient ly by

f in i t e -d i f ference m ethods . At l arg e R r

x

(#-

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V O L U M E 35 , N U M B E R 17

P H Y S I C A L R E V I E W L E T T E R S

27 OCTOBER 1975

The values agree wel l wi th the var ia t ional re

s u l t s ,

- 0 . 2 5 1 9 3 an d -0 . 2 5 0 03 R y , of O 'Mal ley

and Gel tman ,

7

ind icat ing that for Feshbach reso

nances the neglect of the coupl ing term is a good

f i rs t appro x imat ion . The curve denoted by "+"

i s m o re a t t r ac t i v e a t s m a l l R than the - cur ve ,

bu t i t has a poten t i a l barr i e r wi th he igh t 4 .8x 10"

3

Ry and behaves asymptot i cal ly as

2/R

2

above the

n= 2 th resho ld . The potent i a l wel l i s no t a t t ra c

t ive enough to support a bound s tate below the n

= 2

thre sho ld but may be s tron g enough to sup port

a shape resonance above the n= 2 th resh old . To

see that th i s i s indeed so , one can in tegrate the

equat ion inR num erical ly wi th the + poten t i a l to

find the pha se shifts . The se pha se shifts a re

p lo t ted versus energy above the n= 2 th resho ld

in Fig. 2(a) w he re a fast var iat io n of pha se shifts

wi th energy can be obse rved . These re sona nce s

features can also be ident i fied in Fig. 2(b) where

plots of 2* (ft) show local ize d re son anc e be hav ior

n e a r R = 12 as reson anc e energy i s approach ed .

From the computed phase sh i f t s , the resonance

posi t ionE

r

and width r a r e found to be 32 and

2 8 m eV , r e s p ec t i v e l y . Th es e v a l u es a re g re a t e r

than the values E

r

= 18 meV and r = 15 meV ob

t a i ned by th e t h ree - s t a t e c l o s e -co u p l i n g ca l cu l a

t i o n wi t h t went y co r re l a t i o n t e rm s , b u t a r e m o re

accu ra t e t h an t h e t h ree - s t a t e c l o s e -co u p l i n g ca l

cu la t ions wi thou t corre la t ion .

2

The third curve denoted by pd i s comple te ly

repu l s iv e and behave s asymptot i cal ly as 9 .71 / ft

2

.

The channel funct ion for this curve has a large

component of [Z

X

,Z

2

] = [2 , l ] , and may be denoted

a s pd. This channel has the mos t repu l s ive po

ten t i a l among the th ree curves and wi l l no t con

t r ibu te s ign i f i cant ly to any exci ta t ion p rocesses

near the n= 2 th reshold .

Since the exci ta t ion p rocesses occur a t smal l

R (R 4 ), fr om Fig. 1 i t i s evident that the ma in

cont r ibu t ion to the exci ta t ion cross sect ions

com es f rom the + channe l , w i th l i t t l e f rom the

other two. However , bec ause of the p re sen ce of

the poten t i a l ba rr i e r , the + channel i s not open

unt i l the ba rr i e r can be pen et ra te d . In th i s n ar

row energy reg ion , only the - c hanne l is open,

i .e . , the one which can support an infini te num

be r of bound s ta te s . I t has been shown that suc h

a channel wi l l have f in i t e exci ta t ion cross sec

t ions (Is - 2s and Is - 2p ) at the threshold.

41 3

In

t h e \ P s t a t e s s t u d i ed h e re , h o wev e r , t h i s t h re s h

old law only has a very l imited range of appl ica

b i l i ty bec ause the + channel wi l l dominate the ex -

2 . O h

P

H L

I

[

CO

UJ

1.0 k

co

< I

I

a. h

0 I 2 3 4 5 6 7 8

ENERGY ABOVE THRESHOLD l6

3

Ry )

R b o h r )

FIG. 2. (a) Computed ph ase shifts for the + channe l .

(b) Radial wave functions F(R) at various energies in

units of 10"

3

Ry , showing the behavior of reso nan ce .

These cont inuum wave funct ions are norm al ized pe r

uni t energy , wi th energies expressed in a tomic uni t s .

ci ta t ion p rocesses wi th in a few meV.

The cross ing be tween the two curves in F ig . 1

can also be expected in terms of the independent

par t i c l e m odel . I have denoted these two cur ves

b y " + " an d " - , " i m p ly i n g a c l o s e r e s em b l an ce t o

the + and - se r i e s c lass i f i cat ion in t roduced by

C o o p e r , F an o , an d P r a t s

8

in the in terpre ta t ion

of the photoionizat ion data of hel ium doubly ex

c i t ed s t a t e s .

9

They show that the obse rved b roa d

s e r i e s co r re s p o n d s t o t h e i n -p h as e co h e ren t o s

ci l lat ion of the radial motion of the two electrons

which can be denoted s imply as

2svp+2pns

and

the narrow ser i es which can be denoted as 2snp

- 2pns. The cros s ing be tween the + and - cur ves

res u l t s f rom the d i f ference in the types of c or re

lat ion effect implied in the combinat ions 2swp

2pns in the region of small R and of large R.

Let 2snp*

3

P denote a s ing le Sla ter de terminant

as co n s t ru c t ed f ro m s i n g l e -p a r t i c l e o rb i t a l s ;

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V O L U M E 3 5 , N U M B E R 17 PHYSIC L REVIEW LETTERS 27 O C T O B E R 1 9 7 5

then

(2snp+2pns)

1

*

3

P=[2s(rMP(r

2

)y

0110

+ (-l)

s

2s(r^(r

1

)y

l010

\

+[2/>(r>s(r

2

)< y

1010

+( - l ^ W f w f r J ^ o i J

= [ 2 s ( r > / > ( r

2

) + ( - l )

s

^ ( r

2

) s ( r

1

) ] ^

0 1 1 0

wh ere S i s 0 and 1 for s ing le t and t r ip le t , re sp ec

t ive ly , and n / ( r

2

) denotes the rad ia l wave func

t ions . As emphas ize d above , the + or

-

ser i es of

Ref. 8 implies the s ign within each of the square

bra ck et s of Eq . (4) . The refo re ,

2snp+2pns

i sa

+ s e r i e s an d

2snp

-

2pns

i s a-s e r i e s f o r

l

P

a t

s m a l l

R

. The potent i a l cur ve asso cia te d wi th

(2snp

-

2pns)

1

P l i es h igher than that asso cia ted

with {2swp +2pns)

1

P as a resul t of the exis tence

of an extra node near

a

=45 . On the o ther hand ,

a t l a rg e

R

(or l arge r

2

) , the exchange effect is

not importan t and the t e rms wi th the factor ( - l )

s

in Eq. (4) can be neglecte d. I t can be shown that

in the asym ptot i c reg ion , the poten t i a l as so cia t

ed with 2snp + 2pns l i es h igher than that associa t

ed with

2snp

-

2pns

, independ ent of

l

P

o r

3

P .

Thus , for

1

P

9

the + curv e l i es lower a t smal l

R

but h igher a t l arge

R

as compared wi th the-

cu rve , resu l t ing in a cros s ing be tween the + and

- cu rve s . On the o ther hand , s inc e (2snp +2pns)

3

P

i s

a -

s e r i e s a t s m a l l R [because of the factor

( - l )

s

, S=1] , i t l i es ab ove the

(2snp

-

2pnsfP

s e r i e s ( th e + s e r i e s ) a t a l l

R

an d n o cu rv e c ro s s

ing i s expected be tween them, as ev idenced by

actual calcu la t ion .

1 0

I t i s in teres t ing to note that the lowes t Fesh-

bach resonance of

l

P

symmetry in H" cannot be

des ignated as

2s2p

X

P,

so that the lowest doubly

exci ted s ta tes cannot a lways be cons t ructed f rom

the a l lowed lowes t combinat ions of s ing le-par t i

c le s ta te s . Th i s a l so exp lains why the 2s2p

X

P

state of H" was not found in the accurate

1/Z

e x

pans ion calcu la t ion of Drake and Dalgarno

1 1

; in

o ther i soe lect r i c sequences of he l ium the

2s2p

X

P

i s the lowes t Feshbach resonance , bu t in H" th i s

s ta t e beco me s a shape reso nan ce . The lowes t

F es h b ach r e s o n an ce i n t h i s cas e s h o u l d b e d e s

ignated as

(2s2p -2p3s)

1

P.

I t s rad iu s i s peaked

a t

R

30 , in cont ra s t to the shape reson anc e in

Fig . 2 (b ) which i s peaked near

R

12.

+[ 2 ( r > s ( r

2

) +( -

l)

s

2s(r

2

)np(ri)]yioio>

4)

I would l ike to thank Pr ofe sso r A . Da lgarno ,

Pro fes sor U . Fan o , and Dr . G. Victor for th e i r

c r i t i ca l co m m en t s o n t h e m an u s c r i p t .

F o r

a

r e c e n t r e v i e w , s e e R . J . R i s l e y , A . K . E d

w a r d s , a n d R . G e b a l l e , P h y s . R e v . A 9 , 1 1 1 5 ( 1 9 7 4 ) .

2

A

0

Jo T a y l o r a n d P . G , B u r k e , P r o c . P h y s . S o c ,

L o n d o n S2, 3 3 6 ( 1 9 6 7 ) ; P . G . B u r k e , A . J . T a y l o r , a n d

S . O r m o n d e ,

ibid.

92 , 3 4 5 ( 1 9 6 7 ) ; J . M a c e k a n d P

0

G .

B u r k e ibid. 9 2 , 3 5 1 ( 1 9 6 7 ) .

3

Se e J

0

M a ce k , P r o c . P h y s . S o c , L o n d o n ^ 2 , 3 6 5

( 1 9 6 7 ) . A n i n i t i a l a t t e m p t t o o b s e r v e t h i s r e s o n a n c e b y

p h o t o d e t a c h m e n t o f H " w a s no t s u c c e s s f u l ( J . C o o p e r

o f J o i n t I n s t i t u t e f o r L a b o r a t o r y A s t r o p h y s i c s , p r i v a t e

c o m m u n i c a t i o n ) .

4 a

M

0

J . Se a to n an d I . C

0

P e r c i v a l , P r o c . C a m b .

P h i l o s

0

So c . J 53 , 654 (1 9 57 ) .

4 b

M

0

G a i l i t i s a n d R . D a m b u r g , P r o c . R o y . S o c . 8 2 ,

1 9 2 (1 9 63 ) .

5

C . D . L i n , P h y s . R e v . A 1 , 1 9 8 6 ( 1 9 7 4 ). F o r e a r

l i e r w o r k , s e e J . M a c e k , J . P h y s . B : A t . M o l . P h y s .

1 . , 8 3 1 (1 9 68 ) ; U. Fan o , in Atomic Physics:

Proceed

ings, e d i t e d b y B . B e d e r s o n , V . W . C o h e n , a n d F . M .

P i c h a n i c k ( P l e n u m , N e w Y o r k , 1 9 6 9 ) , V o l . I , p . 2 0 9 .

T h e a c t u a l m e t h o d of c a l c u l a t i o n u s e d h e r e i s d i f f e r e n t

f r o m t h a t e m p l o y e d b y M a c e k a n d F a n o ;ac o m p l e t e d i s

c u s s i o n w i t h c o m p a r i s o n s w i l l b e r e p o r t e d e l s e w h e r e .

6

M a c e k , R e f . 5 .

7

T , F

0

O ' M a l l e y a n d S . G e l t m a n , P h y s . R e v . 1 3 7 ,

A1 3 44 (1 9 65 ) .

8

J . W . C o o p e r , U . F a n o , a n d F . P r a t s , P h y s . R e v .

L e t t . 1 0 , 51 8 (1 9 63 ) .

9

R . P . M a d d e n a n d K . C o d l i n g , P h y s . R e v . L e t t . 1 ,

51 6 (1 9 63 ) .

10

F o r

3

P o f H " , c a l c u l a t i o n s s h o w t h a t t h e c u r v e l a

b e l e d " " i s c o m p l e t e l y r e p u l s i v e . T h e r e s u l t s a r e t o

b e r e p o r t e d e l s e w h e r e . F o r

3

P o f H e , t h e c o m p u t e d

c u r v e s d o n o t h a v e a n y c r o s s i n g ; s e e M a c e k , R e f . 5 .

1

G . W . F . D r a k e a n d A . D a l g a r n o , P r o c . R o y . S o c ,

L o n d o n , S e r . A 3 2 , 5 4 9 ( 1 9 7 1 ). S e e a l s o , D . R . H e r -

r i c k a n d O . S i n g a n o g l u , P h y s . R e v . A 1 1 , 9 7 ( 1 9 7 5 ) .

1153