Recurrence relations for quantal multipole radial integrals

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Recurrence relations for quantal multipole radial integrals

N. Cherkaoui, J. Chapelle, C. de Izarra, O. Motapon, J. Picart, N. Tran Minh, O. Vallée

To cite this version:

N. Cherkaoui, J. Chapelle, C. de Izarra, O. Motapon, J. Picart, et al.. Recurrence relations for quantal multipole radial integrals. Journal de Physique II, EDP Sciences, 1994, 4 (4), pp.573-579.

�10.1051/jp2:1994148�. �jpa-00247983�

Classification

Physic-s

Abstracts 32.70C

Recurrence relations for quantal multipole radial integrals

N. Cherkaoui

iii.

J.

Chapelle ('),

C. de Izarra

(~),

O.

Motapon (~),

J. Picart

(~),

N. Tran Minh (~) ^{and O. Vallde}

(1)

(')

GREMI, Universit6 d'orldans, B-P. 6759, 45067 Orldans Cedex, France (h

DAMAp

(*), Observatoire de Paris-Meudon, 92195 Meudon Cedex, France

(Receii>ed 7 October J993, ieceii,ed in final

form

20 December J993,

accepted

3 Januarv J994)

Rdsumd. Dans

un pr6c6dent article, _nous avonslnontr6 dans le cadre de

l'approximation

coulombienne

semi-classique

(WKB) _que les int6grales radiales des transitions multipolaires entre les dtats

atomiques

_non hydrog6noides ob6issent h des relations de r6currence simples. Dans _cet

article, _nous 6tablissons les relations de rdcurrence reliantles

int6grales

radiates _non

hydrogdnoides

exprim6es _sous la forme

quantique.

Les tests

num6riques

effectu6s donnent de bons rdsultats.

Abstract. In _a previous paper, we showed that in the semiclassical (WKB) Coulomb

approximation, the radial integrals for

mu)tipole

transitions between

nonhydrogenic

atomic _states obey

simple

_recurrence relations. This paper deals with the _recurrence relations

connecting

the

nonhydrogenic

radial matrix elements

expressed

in the quantal form. Numerical tests have been made and

give

good results.

1. Introduction.

Many

recent studies in laser spectroscopy cannot be

interpreted

if _one does not know the

matrix elements of the operator

r~,

for both

hydrogenic

and

non-hydrogenic

atoms, and for _a wide range of the quantum ^numbers u, I _and v',

I'.

_Such

a case occurs when _one studies the

effects of _a laser radiation _on atomic

spectral

lines. These effects have been studied and

explained by

Dubreuil et al.

[1-3]

in their articles that illustrate

clearly

the use of the radial

integrals

in bound-bound and bound-free transitions.

The electric

dipole

transitions, that have _a dominant role in the

interpretation

of

experimental

results, have

given

rise, _more than the transitions of

higher

orders, _to the interest of

physicists. Nevertheless,

for

hydrogenic

_{cases v} and v'

integers

_a ^few studies

[4-6]

yield

the

analytical expressions

of the matrix elements

(nilr~/n'i'). _Recently,

Price and Harmin

[7, 8],

and also Braun

[9]

have used _a differential

equation

to calculate the

hydrogenic dipole

matrix elements.

Blanchard[6]

has found _a

general

recurrence relation

linking

(*1URA 812.

574 JOURNAL DE

PHYSIQUE

II N° 4

(nilr~

+

(n' I')

_to

(nilr~/n I'), (nilrfl

^' In'

I')

and

(nilrfl ~/n' I')

for any

complex

number p

satisfying

Re

(fl )

_m

I I'.

_{In the}

particular

_case where _n

=

n',

he

simply

obtains

a

proportionality

relation between

(nilr~

^~

~/n' I')

and

(nilr~ ~'In' I')

for any L

verifying

ji I'[

_< L _w

I

+

I'.

_{Similar relations}

were obtained

by Epstein

et al.

[10],

and also,

by

the

use of

hypervirial

relations

by Hughes

I

il. Ojha

et al.

[12]

^have demonstrated the relations of Blanchard,

using

^the classical

properties

of the

generalized hypergeometric

functions _; and

Swainson et al.

[13j

ha,<e

given

an alternative

proof

for these relations.

However, theoreticians in different branches of atomic

physics

_are still interested in the

analysis

of the

hydrogenic

matrix elements,

particular

attention has been

paid during

the last years ^to the

non-hydrogenic

matrix elements

[14-16, 201.

In the

majority

of the articles, the theoretical

study

of the radial

integrals

is made in the Coulomb

approximation

that facilitates the mathematical _treatment of the

Schr6dinger equation.

Up

to now, no recurrence relation between the

non-hydrogenic

matrix elements

expressed

in

the

quantal

form of the Coulomb

approximation

has been

published.

On the other hand, such relations have been obtained in the semiclassical _case

[20]. Therefore,

_we want in this paper ^to establish _recurrence relations

connecting

the

non-hydrogenic

radial

multipole integrals

in the

quantal

_case.

2. Relations between

non.hydrogenic

matrix elements.

In the

quantal theory,

the radial

integral

for _a

multipole

transition reads

[16]

l~

R~

₌

(vi [~[ v'i')

=

drP~~ i~P~

~,

(l)

o

where L is the order of the

multipole

(L

=

I for _a

dipole radiation)

and the P's _are the

wavefunctions of the two states involved with

energies

E

=

and E'=

2 v

~ 2

v'~

respectively.

In the Coulomb

approximation,

the radial

integral

is taken from the cut-off radius

1-~ and the wavefunctions _can be

replaced by

their

asymptotic

forms, which _are

purely

Coulomb solutions,

P _pi K

w,.

+ in (>.) x ₌

~~ (2)

and

similarly

for

P~

_~,. Here, W~~ ^denotes a Whittaker function

[17],

which vanishes

exponentially

_{as x}

- co, but is

irregular

at the

origin

for _non

integer

_u, For the normalization

constant K, _{we use} the Hartree

generalized hydrogenic

form, I-e,

~

~

j~~ r~~ ~~, _r~~

_{~ ~}

~~j~i,~

_~~~

A _more refined normalization is

required only

^for low values of _v,

especially

when

v ~

i

₊₁

[16, 21]. Thus,

the radial

integral

of

equation (I)

_can be written

RL ₌ ^K K' dr

Wp,

_{I +1<2}

~) ^~ _Wp,,

_I,

~ in

)I ₍₄₎

Putting

k

=

I'-

I,

_one

can rewrite

(4)

in the form

R~

₌ ^{K K'}

I~

,

(5)

where

lL,

_k "

ld~

_~i~v,

f+ f<2

~/ ^~

~i~v'.f

+ i ₊ f<2

)~ ⁽⁶⁾

~

If _one takes _:

~ '~~'~ "

i'~'~

W,.,

+ j,~

~ _i'

v

(7)

and

V2(I)

_~ ^l'~'~

W,",

_I

+i +1/2

)~

~

V2,1(I) (8)

the _two functions

Vi (r)

and

V~(r) satisfy

the differential

equations [18]

_:

~

-i(I+1)+~ (~+ i)

~.~~'_~~~'+~.

l

+~+

^{~ ~}

vj(r)=o (9a)

dr2

dr v2 _r

i~

and

~2~

_~~

(k

₊ ^I

)(k

₊

I

_{+ 1)}

+

~

+ l

r

fi

_L

~

+ r +

~ +

~

~

V~(r)

=

0

(9b)

dr2

dr

v

'2 r r-

The

multiplication

of the first and second

equations by Vz(r)

^and

Vi (r) respectively,

followed

by

substraction and

subsequent integration

between r~ and _co leads _to

~°

_i i

I(I

₊ I)

(k+I)(k+I

₊

i)j

~

~_~~

~,~ ^~~_~

,

v~

^~

v'~ r~

^~ r~ ^{' ~}

°° ^d~vi d~v~

j°°

dvj dv~

+ r

v~ vi

^dr L

v~

v,

dr

=

o.

lo)

,~

dr~ dr~

,~

dr dr

After

integration by

_parts, one arrives at

(-(+() _i~~j,,+ i(k+ii(k+I+ i)~i(I+ i)i.i~~j_,+

dV dV~ JJ

~°

dV

~

+

(r ^V~

_dr ^V _di^{/ +}

^{L) V,}

^{V~ + 2(1 + L) V~ dr = 0. (I I}

,~ ,~

dr

With

regard

to the difference

equations

which _are satisfied

by

the

V~

^{function with} respect ^to the parameter k, _we mention the two formulae

[19j

dV~ ~~~~~)

i i

$~

r

~i+k ^~~ ii+k~v'~~~"~'

~~~~~

576 JOURNAL DE PHYSIQUE II N° 4

Using

these _two formulae in

equation

I I), the

following

_recurrence

relations,

satisfied

by

the

integrals J~,,

are

easily

obtained

AJ~~j,+aJ~,+bJ~,_~ +cI~_j,

+d=0

(13a)

and

A/~~,,+a'/~,+b'/~,~j

+c

/~_,,+d=o. (13b)

Subtracting (13b)

from

(13a),

_one has

(a

-a')/~,

+

b/~,_j b'/~,~,

+

(c c')/~_,

, ⁼

0

(13c)

where

A

=

~

+

~

(14)

v- v'-

~

i+k

^~

i+k+1

^~~~~

b-21L+')1]+11. h2(L+i)l~+I+i-)1 ₍₁₆₎

c _t

ik(i+ i)+L(L+1)+2ik~2(I+ki(L+ i)1 (17)

c'=

[k(k+ I)+L(L+1)+2ik+2(I+k+ I)(L+1)] (18)

dv, dv~

C~

d

=

(r (v~

d~

^vi

dl'

^(I

⁺

^{L) vj} v~j ⁽¹⁹⁾

,~

It is obvious, _as in the semiclassical _recurrence relations

[20],

that _some

integrals

in relations

(13a), (13b)

and

(13c)

_are not

physical

since

they

_are forbidden

by

the selection rules.

3. Numerical calculations.

Since the

quantal

recurrence relations _are not

analytic

and

they

contain

non-physical integrals,

it is necessary ^to test them

numerically.

We used Klarsfeld's B method _to compute ^the

integrals

I~_ _~~. The numerical values of the constant A, _a, a', b, b', _c and c' _are

easily

obtained whenever the quantum ^{numbers of the transition} are chosen. The

problem

in the _test of the _recurrence

relations (13a) and

(13b)

_comes from the d constant. In relation

(13c)

d has been eliminated.

3.I DETERMINATION _{oF THE} d _CONSTANT.

Equations

(7),

(8)

and

(19)

lead to

~

~

[l~

^~

^~i~v',

^I + i +1<2

)~ ) _W;,

I _+1<2

~/

~Wv,i~l12~ ~) )Wv',I+i+1<2( )~

(l +L)i~W~,~~j~j~(

~~

W~,,~~,~j,z( ~( ^~~° _{=d~-d, (20)}

v v ,~ ^~

where

d~

is the value at

infinity

and d,~ ^{that at} r~. The Whittaker functions and their derivatives

are known _to vanish _as the argument tends to

infinity

_;

therefore, d~

₌ 0. Then d

=

d, (21)

1-o lo ^~ 2.0

10'~

O-O10°

1-O

10'~

1,510'~

2.o lo ^~ l.0 lo ^~

3 o

i~-3

4,0

lo'~

^5.010

~

5.o

io'~

_o-o

io°

7 9 11 13 15 17 19 21 23 _lo 15 20 25 30 35 40

Fig. I. Fig. 2.

Fig. I. Graph of f(r~) ₌ d/I~_

~ i>ersus r~, v = 18.90, v' 20.50, f 2, f' I, L 2

Fig. 2.

Graph

of f(r~ j d/J~_ _i~ersus i~. v = 29.10, v' 30.90, ^f 3, f' 4, L 4.

d,~

depends

on the quantum numbers, and

closely

_{on r~. r~} has not _a fixed value and varies

between authors.

According

to Bates and

Damgaard [21j

i~ = a

~~

,,

with 1~ _a

~ ?.

(22)

v +

Plotting d/JL

i ^"

f (r~),

where

IL

i is the smallest

integral

of relations

(13a)

and

(13b),

one

realizes,

_{as can} be _seen in

figures

I and 2, and in the range of r~ ^{we are} interested in that _: I) For transitions with

high

quantum numbers, d _can be taken

equal

to zero, because of the number of

significant digits

which is 4 (see

Fig.

2).

ii)

For transitions with moderate _or small quantum

numbers,

the absolute value of

f'(r~

^is

always

less than

0.005,

and

f (i~

is random d _can then be taken

equal

to zero, and the

hypothesis

of this choice will be verified in the test of relations

(13a)

and

(13b).

3.2 NUMERICAL _{TESTS oF THE RECURRENCE RELATIONS AND} DIscussioN. In table I that

follows, _we present a few transitions and the

corresponding

values of the

integrals I~

,. For any transition, _we calculate the value of the

integral /~

~, , from relation

(13a)

TESTI

-,

of _course

putting

d

=

0

comparing

it to the value obtained

by

direct calculation,

we have the relative _error El.

Similarly,

TEST2 and E2 _are the value of

/~

~ ~_, calculated from

(13b)

and the associated relative _error. TEST3 compares the values of the

integral /~

, obtained from relation

(13c)

and

by

direct

calculation,

and E3 is the relative _error

yielded by

this

comparison.

It _comes from table I, that, in

using

the derived

quantal

recurrence relations, one can obtain

radial

integrals

in

good

agreement ^{with those}

yielded by

direct calculations. TEST3, which

uses

equation (13c), gives

_exact results for

/~

, while TEST I and TEST2, that

depend

on the

d _constant involve _errors that _can be

explained by

^the random character of d. Furthermore, the

smallness of the _errors less than I §b account for the choice of d taken _to be _zero and constitutes _a

good

test for _{our recurrence} relations. These relations _are stable because their

repeated application

does _not involve _a loss of

precision.

4. Conclusion.

In a

previous

_paper Picart _et al.

[20j

_we have shown that

dipole quadrupole

and

octupole

radial matrix

elements,

calculated

using

the semiclassical

(WKB)

form of the

578 JOURNAL DE PHYSIQUE II N° 4

22

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Coulomb

approximation, obey simple

_recurrence relations. Here, _we have established

recurrence relations between

multipole

radial matrix elements obtained from the quantum

mechanical form of the Coulomb

approximation.

These relations have been tested and the numerical results _are

proof

of their

validity.

A

relatively large

^{number of the radial matrix} elements _can be obtained from _a few _ones

although

_some of them _are not

physical.

References

ii Dubreuil B., Chapelle J., J Phi.<. Collo~ Fiafice 34 (1973) C2-8.

[2j Dubreuil B.,

Chapelle

J.,

Phys.

Lett 46A (1974) 451.

[3] Dubreuil B.,

Chapelle

J., Phj,sic.a 94C (1978) 233.

[4j Bessis N., Bessis G.,

Hadinger

G., Phy.v. ^{Rei~. A28} (1973) 2246.

[5] Badawi M., Bessis N., Bessis G., Hadinger G., Phys. Rei~. AS (1973j 727.

[6] Blanchard P., J. Phys. B.. At. Mol. Phvs. 7 (1974) 993.

[7] Price P. N., Harmin D. A., Ph_vs. Rev A42 (1990) 2555.

[8] Price P. N., Harmin D. A., Phj°s. Rev. A46 (1992) 6110.

[9] Braun P. A., Phy.v. Rei~. A46 (1992) 6108.

[10]

Epstein

S. T., Epstein J. H., Kennedy B., I. Math. Phy.v. 8 (1967) 1747.

[1II Hughes D. E., J Phys. B10 (19771 3167.

[12]

Ojha

P. C., Crothers D. S. F., J. Phv.v. B At Mol. Pit,'s. 17 (1984) 4797.

[13) Swainson R. A., Drake G. W. F., I. Phvs. B.. At. Mol. Opt. Phys 23 (1990) 1079.

[14] Picart J., de Izarra C., Oumarou B., Tran Minh N., Klar~feld S., J. Phj~s. B. At. Mol. Opt. Pliys 23 (1990) L61.

[15] Oumarou B., Thbse de doctorat, Orldans (19861.

[16] Klarsfeld S., Phj's. Reii. A39 (1989) 2324.

[17] Whittaker E. T., Watson G. N., A _course of Modern Analysi~ (Cambridge University Press, Cambridge, 19521.

[18] Handbook of mathematical function~, M. Abramowitz and I. A. Stegun Ed~. (Dover, New York, 1965j.

[19] Buchholz H., The Confluent

Hypergeometric

Function, vol. Is

(Springer

Tracts in Natural

Philosophy,

1969).

[20] Picari J., de Izarra C., Oumarou B., Tran Minh N., Klarsfeld S., Ph_v.v. Rev A43 (1991) 2535.

[2 ii Bates D. R. and

Damgaard

A., Philos Tians. R Sac- A242 (1949) ^101.