Atomic parameters for transitions involving Rydberg states of singly ionized alkaline earths

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Atomic parameters for transitions involving Rydberg states of singly ionized alkaline earths

Mohamed Djerad

To cite this version:

Mohamed Djerad. Atomic parameters for transitions involving Rydberg states of singly ionized alkaline

earths. Journal de Physique II, EDP Sciences, 1991, 1 (1), pp.1-9. �10.1051/jp2:1991135�. �jpa-

00247496�

J.

Phys.

II1

(1991)

1-9 JANVIER1991, PAGE

Classification

Physics

Abstracts

32.70C

Atomic parameters ^for transitions involving Rydberg states

of singly ionized alkaline earths

Mohamed Tahar

Djerad

Dkpartement

de

Physi#ue,

Facultk des Sciences, 1060 Tunis, Tunisia

(Received

it

July,

1990,

accepted

8 October,

1990)

Rksumk. En utilisant des fonctions d'onde de la thkorie du dkfaut

quantique,

nous avons

caiculk les kikments de matrice radiaux mettant en jeu les ktats de

Rydberg

des atomes alcalineux

terreux une fois ionisks. Dans le but de tester les rksultats, _nous avons calculk les

probabilitks

de

transitions, ies forces d'oscillateurs et ies durkes de vie radiative des ktats S, P, D et F

jusqu'i

n ~

30

( *).

Les

probabilitks

de. transitions sont en bon accord _avec ies rksultats

disponibles.

Nous

prksentons

les durkes de vie eh terrre de lois d'kcheiles du type _r~i

=

ro(n*)~.

La variation des forces d'oscillateurs _en fonction du nombre

quantique

effectif n* est discutke.

Abstract.

Using quantum-defect theory

wave functions, _we have computed

dipole

^radial matrix elements

involving Rydberg

states of the

singly-ionized

alkaline-earth atoms. In order. to test the results, _we have calculated a set of transition

probabilities, absorption

oscillator strengths and radiative lifetimes for S, P, D, and F states up ^to n 30(^**). Transition

probabilities

are in

good agreement

_with available data. We report ^{the lifetimes in} terms of scaling relations of type r~i =

r~(n*)~.

The variation of oscillator

strengths

with respect to the effective

principal quantun~

number n* is discussed.

1. InUoducdon.

Study

of the

properties

of atomic

Rydberg

states is of current interest. Excited atomic states are

pertinent

to

applications

in diverse fields such _as

astrophysics, plasma physics,

space

physics,

_etc.

Knowledge

of transition

probabilities,

oscillator

strengths

and radiative lifetimes is often needed. When

available,

measurements of these parameters are

generally

^limited to

low-lying

states.

Thus,

_a theoretical treatment

including highly

excited-states is of interest.

The calculation of any of these

parameters requires dipole

radial matrix elements. Extensive

calculations, involving

the _use of elaborate theoretical model

requires prolonged computation

(*)

L'ensemble des rksultats est

disponible

_sur demande adresske I l'auteur.

(**)

These extensive data _are available _upon request from the author.

2 JOURNAL DE

PHYSIQUE

II M

times.

Therefore, only approximate

^methods are

usually

considered. The main

problem

is to

develop

_a method for

obtaining satisfactory

and convenient _wave functions for the

highly

excited-states. The traditional _way of

describing Rydberg-state

_wave ^{functions is}

bj uie

of the

quantum-defect theory,

which _was

inaugurated by

Bates and

Damagaard

in their classic paper

[I]. Using

a numerical

approach

to

Bates-Damagaard

method

Lindgard

and Nielsen

[2]

evaluated transition

probabilities

for the alkali isoelectronic _sequences for states with

n w12. To _our

knowledge,

the atomic parameters are not

presently

available for

high-n-

excited states of

quasihydrogenic systems,

except alkali _atoms for which exist extensive

predictions [3].

We consider

singly-ionized

alkaline-earths. We have used

analytical

wavefunc- tions

[4],

^which were shown to

provide

_a

good compromise

for detailed atomic calculations and _accuracy

[5].

The paper is

organised

_as follows. In section

2,

we

briefly

summarise the method. In section

3,

we determine the

parameters required

for the

computations.

In the last

section,

_we

comment on the results. All the results _are not listed here but _are available _upon request from

the author. Atomic units _are used

throughout

unless otherwise stated

explicitly,

2. Method.

Kostelecky

_et al.

[4] proposed

_an

exactly

solvable

potential

model

adapted

to the

problem

of one-electron atoms. We

briefly

summarise the main feature of the model and

give

the

working equations.

The outer electron is described

by

the

following

orthonormalized radial

wave

functions, expressed

in _terms of _gamma

(r)

^functions

~

~~~~

ⁱ

P3r(n*

/* I

(/))

^>'2

fir

j'.~-prj~n. ~ji,~>~, ^fir

~"'"

@

2

r(n*

₊ I* ₊ I

)

^{n* ~ n* '}

where

p

₌ 2

z~,

_~~ is the net

charge

of the

nucleus,

I*

= 3

~j +

I(1), _3~j

is the quantum defect of _a state nl

)

and

I(I )

is _a

phenomenological, non-negative integer

with the allowed range of variation determined

by

the

following

_two conditions _:

(a)

the kinetic and

potential energies

_are

separately normalisable,

which

imposes

the constraint 3

(1)

-1-

<

I(I (b)

2

Laguerre polynomials L)I '"it'~,

exist

only

for

I(I)

_{w n~,~} l

I,

where _{n~~~ is} the

principal

quantum number of the

ground

state.

For _a transition from the level nj

ii

to the level n~

l~),

the radial

part

^of the

dipole

matrix element is

given by

(n,*)~" (nf)~~~

Jo ~ q~4

(n~ _/~(

r ni

/,)

= ~

F(I, f)

_x

4

z~(nj* nf

x

(r(n,* /~*) r(n? It r(n~*

+ /~* + i

) r(n?

₊

It

₊

1))"2

~

p~jo ~o

PI q'

~

(~l

~~~~~~) ~ n/~~~~~~j

~~~~i*~~ /~)ll'~

q + 2 ^'

where

F(I,f) =2ni*nf/(n~*+nf).

These matrix elements _are

completely analytic

and

therefore convenient for calculations. In the

Bates-Damagaard [I] model,

the radial matrix elements _are obtained

by

numerical

integration.

Moreover,

they depend

_{on a} cutoff radius.

Various research _groups have been

implementing

various series cutoff criteria but _no

precise

justification

^{could be}

^given.

M I RYDBERG STATES OF IONIZED ALKALINE EARTHS 3

3.

Applicadon

to

singly-ionized

alkaline-earth atoms.

Einstein

coefficients,

oscillator

strengths

and lifetimes have been calculated for several

quasihydrogenic

atoms

(ions).

In

general,

the calculations _were limited to

low-lying

states of the

systems.

^{In the}

present work,

_we include transitions

involving Rydberg

states of the second

spectra

^{of the alkaline earths. For} a

given ion,

the basic

parameters

^{of the method} will

now be

specified.

3.I THE QUANTUM DEFECT. It is well known that the

singly-excited binding energies

of

monovalent atoms

(ions)

in state

(n/)

_are

given by hydrogen-like

formula

~

Zj Zj

~~

2(n*

_)~

2(n

3

~j)~

For _a

particular angular

momentum

/,

the

quantum

^defect &~i is

nearly independent

of

n. To evaluate the

3~i

^of

high

_n states and thus

E~j,

we have used _a numerical fit and

employed

the extended Ritz formula _:

&~,=a+hm~~+cm~~+dm~~

_with

_m=n-30.

For each nl

series, 30

is constant and its value is determined

according

to _an

optimisation technique.

The initial value is. the

quantum

defect of the lowest ternJ. In most cases,

30

could be obtained

by rounding

the initial value to the nearest 0.I. As recommended

by

Martin

[6],

the present Ritz formula is more convenient than _one

expressed

in powers of

n*,

which has been

frequently.

As

input data,

_we have used available

experimental

excitation

energies [7] only experimental

data for lower levels have been considered. Since _we

neglect

the fine structure of the

levels,

we have used the baricenter for each level associated with the two fine-structure

components.

In all _cases, the fit

reproduces

the observed

energies

to better than I fb. For

instance,

for Ba

ion,

there exist _accurate

experimental energies [8]

_up to

n = 70 for _{ns, np,}

nd,

and _ng

Rydberg

series. We have verified that _we

reproduce

these values with excellent agreement. ^{We have listed the fit} parameters ^{in table1.}

3.2 THE _INTEGER

I(I). Recently, Kostelecky

etal.

[9] predicted

^the fine-structure

splittings

for

quasihydrogenic

atoms

(ions).

These _are relevant to the short-distance behaviour because

they

_{vary as} r~ and r~ ^~ and

depend

_on the

integer

I

(I )

in the model.

First,

the authors of reference

[9]

showed that the model constitutes _a

good approximation

for

large

r and is therefore convenient for

Rydberg

states.

Second,

the model

provides good

predictions

of the fine-structure

splittings

for

light

atoms.

Accordingly,

the

experimental

data for fine-structure

splittings

allow the determination of values of the

phenomenological integer I(/ ).

In our case, this

approach

works well for Be⁺

only.

For the other

ions,

we have selected the values of

I(/)

that

give

the best agreement ^with

accepted

values. Table II shows the

phenomenological integers

_we have used for various values off and for ions studied

by

_us.

4. Results and accuracy.

We have

performed

calculations _{on a} Vax 785 computer. ^The

expectation

values of radial

integrals

_are

expressed

in terms of r function

(see

Sect.

2).

The _use of double

precision

limited the calculations to n ₌ 15.

Quadrupole precision

_was therefore

used, corresponding

to about 33 decimal

digits.

The radial wavefunction

depends

_upon the energy

E~,

of its

corresponding

state and the

4 JOURNAL DE

PHYSIQUE

II N

Table I. Excitation

energies of

the

fit

parameters.

Ion

3~

a h _c d

s 0.3 0.25826767 0.072383024 0.27282557 0.49940956

Be+ p

0.0'

0.049366057 0.016608201 0.30420470 0.73083270

d 0.0

0.00071538i12

0.062185451 0.97974998 4.8120747

f 0.0 0.00029005721 0.0051964801 0.0733184467 0.0

s

1[0 J.066)201

0.13960186

0.2898719§

0.91979468

p 1.0 0.69639760 0.14831746 0.020831158 0.0

d 0.6 0.045457553 0.094118074 0.024890661 0.33961356

f 0.0 0.0035436391 0.037342068 0.73627400 6.6747470

g 0.0 0.0037441782

0.40812709

_{40.203358 075.3479}

s 2.0 1.7994976 0.29482222 0.67478645 1.7830194

p 1.9 1.4364885 0.29501835 0.0 0.0

Ca+ d 1.0 0.62281942 0.21231084 0.96863210 4.6014581

f 0.0 0.025947075 0.15336217 4.4063873 61.276630

g 0.0 0.006863834 1.1933397 43.406574 507.34344

s 3.0 2.7050006 0.30806586

0.278495f4

0.97451008

p 3.45 2.3297160 0.16261426 0.0 0.0

Sr+ d 1.40 1.4505944 0.51244223 2,1281478 >27.432442

f 0.0 0.061722245 0.19285420 5.8515310 48.552032

g 0.0 0.98785323 £ 0.28638694 7.2725739 64.667908

s 4.0 3.5765221 0.36527434 0.15957843 0.62172544

p 3.95 3.191788 0.37837729 0.0 0.0

Ba+ d 2.40 2.3716066 0.91275573 4.928298 60.386803

f 3.0 0.85638016 0.73997784 0.58487803 0.78243428

g 3.4 0.020221187 0.040776726 0.13081138 0.16473958

ionization

potential

of the ion. It is obvious that the results

depend

_upon the _accuracy of the

experimental

level data.

Obtaining

these data from the extended Ritz formula eliminates local

accidents.

In

assessing

the accuracy of atomic parameters ^{it is} _necessary to consider whether the radial

integral

is sensitive to small

changes

in the _wave functions. The

sensitivity

is

usually

determined

by

the

degree

of cancellation which _occurs between

positive

and

negative

contributions to the radial

integral.

In order to obtain _an indication about such

sensitivity

let _{us assume} that all the excitation

energies

_are determined within I fb this

corresponds roughly

to ad

uncertainty

^An the

quantum

^{defect 0.03} w ha

~j w 0.30 for the studied _range of

n. We have varied

arbitrarily

the

quantum

^{defects for initial} or final states in the aforesaid _range. We have observed that the radial

integrals

_are stable within

roughly

I fb for all transitions. Of _course, if we,take into

account final and initial states, the cancellation effects would have

larger

percentages.

The transition

probability

for

spontaneous

emission from the upper level _(n~

l~)

^to a lower

M I RYDBERG STATES OF IONIZED ALKALINE EARTHS 5

Table II. Values

of

the parameters

I(I),

_a and _{To iR} 10~ ^~ _s

(see

the text

)

and the

asymptotic

quantum

defects 3,(1) for

values

of

the quantum number I.

I Parameter Be+

Mg+

Ca+ Sr+ Ba+

1(0)

0 2 4 4

s a 2.94 2.95 2.91 3.02 2.95

To 5.97 9,19 0.13 0.06 0.07

3(0)

0.258 1.066 1.800 2.705 3.577

1(1)

0 2 2 3

p a 2.92 2.96 2.89 2.87 3.08

T~ 9.53 0.239 0.27 1.0 0.80

3(1)

0.049 0.697 1.437 2.329 3.192

1(2)

0 0 2

d a 2.97 3.00 3.00 2.93 2.93

To 3.05 0.156 7.90 0.073 0.884

3(2)

0.001 0.045 0.623 1.451 2.373

1(3)

0 0 0 0 0

f _a 2.97 3.02 2.93 3.17

To 6.50 0.528 0.055 0.320

3(3)

0.000 0.004 0.026 0.062 0.855

g

1(4)

0 0 0 0

3(4)

0.000 0.003 0.006 0.988 0.020

level

(n~l~)

^{for excitation}

^energies E~,j,

^and _E~~j~,

respectively,

is

given

in units of

l0~s~'by

2.026 _x

10'~ (~i ~

_~

(~f~f~ _'~

~~~ ~~~'~~~

A

(ni

_{l~ ~}

n~l~)

=

~ ^~ 2

(

₊ ^'

where A

(in A)

is the

wavelength

of the transition involved. The line

strength

(n~

l~( r

(n~l~)

_{(~ is} in atomic units. In

tablet @IIa), (IIIb)

^and

(IIIC),

_we list _{a set} of _our calculations fur the transition

probabilities

for

light,

medium and

heavy ions, namely, Be+,

Ca+ and

Ba+, respectively.

The results _are

compared

with recent critical

compi- lation[10].

The

majority

of the

quoted

values _are obtained

calculationally

rather than

experimentally. Typically,

these values _are

expected

_to be accurate to within about 25 fb in

the Be+ and Ca+ _cases and within about 50 fb in the Ba+ _case.

However,

_we note certain poverty ^for ns _~ np and np ~ ns transitions. For

example

the coefficient for Be +

(4s

_~

2p)

transition is 1.5 greater ^{than the}

accepted

_one. This is

expected

because these transitions involve the

largest quantum

defects and should

provide

the most

stringent

test of the model.- Moreover the model does not take into _account

polarisation

effects. The results

conceming

the other levels _are in the error bars and in certain _cases to

6 JOURNAL DE PHYSIQUE II lK°

within _a fraction of

percent.

This is due to the fact

thjt

these levels _are almost

hydrogenic,

even for the haviest ions. We

emphasize

that within the

present'model,

_we model less

accurately low-lying

levels than those with

larger

values of _n which _are of

primary

interest in the

present

calculations.

Thus,

the

agreement

^with

accepted

values _may be _{seen as} better than that shown in tables III.

Table III.

Comparison of

our transition

probabilities

with

accepted

values

for

the ions

of Be(IIIa), Ca(IIIb)

and

Ba(IIIC).

(a)

Transition Present

Accepted

(io8

_~->)

(io8 ~-l)

6s ~

4p

0.146 0.102

5s

~

3p

^{0Al1 0.28}

4s ~

2p

0.142 0.94

6p

_~ 4s, 0.0228 0.0256

5p

~ 4s 0.0259 0.030

5p

~ 3s 0.107 0.142

4p

_~ 3s 0.143 0.19

6d _~

4p

0.137 0.143

5d

~

3p

0.576 0.59

4d ~

3p

1.13 1-1

3d ~

2p

11.6 9.2

(b)

Transition Present

Accepted

(jo8

_~->)

(jo8 _~->)

7s ~

4p

0.201 0.308

6s ~

4p

0A13 0.62

5s

~

4p

1.17 1.7

4p

_~ 4s 1.32 1.47

7d ~

4p

0.227 0.265

6d ~

4p

0.422 0.354

5d _~

4p

0.968 0.97

4d ~

4p

3.62 3.6

lK° RYDBERG STATES OF IONIZED ALKALINE EARTHS

Table

III(c).

Transition Present

Accepted

(108 s-') ₍₁₀₈ s-')

I ls

~

8p

0.0456 0.041

1Is ~

7p

0.0744 0.064

II _s

~

6p

0.160 0.12

10s

~

6p

0.250 0.18

9s

~

7p

0.218 0.18

9s

~

6p

0.428 0.32

8s ~

6p

0.840 0.60

8p

~ 7s 0.0177 0.018

7p

~ 7s 0.181 0.20

7p

~ 6s 0.119 0.10

6p

~ 6s 1.04 1-11

8p

~ 6d 2.18 _x 10-4 6.7 _x 10-4

8p

_~ 5d 0.0176 0.034

6p

_~ 5d 0.176 0.332

10d

~

8p

0.0340 0.037

10d

~

7p

0.0584 0.058

10d ~

6p

0.138 0.13

9d

~

8p

0.0562 0.063

9d

~

7p

0.0929 0.093

9d

~

6p

^{0.214 0.20}

8d ~

8p

0. I I1 0.13

8d ~

7p

0.164 0.16

8d ~

6p

0.360 0.40

7d ~

7p

0.361 0.44

6d

~

6p

^{1.77 1.80}

9d ~ 5f 4.63 _x 10-4 6.6 _x 10-4

10f ~ 5d 0.0106 0.038

6f

~ 5d 0.124 0.22

The

absorption

oscillator

strength

is

given by

the dimensionless

quantity

303_75

(n~ lj

_r

(n~l~)

^~ max

((, l~)

~~~~ ~'~

~~~~~

A 2 _{l~ +}

It has

long

been established that

spectral

series of

quasihydrogenic species

reveal

systematic

trends. A

relatively unique

aspect ^of

Rydberg

series is their

regularity. They provide

_an

opportunity

for-

systematic

and self-consistent _studies

by

virtue of the

n-dependence

of

8 JOURNAL DE

PHYSIQUE

II lK°

it

id

~f

~~.1

0 _IO 20 30

n~

Fig.

^I.

Systematic

trend off values witlfin the

Mg+ (nS

_-

4P) (x)

and Sr+

(nS

- 5P)

(A~

series.

Plotted 18

(n*)~

f _vs., the effective

principal

_quantum number n*.

/ ,

virtually

all atomic

proprieties.

For

instance,

the oscillator

strengths

diminish _as

(n*)~~

_so

that

(n*)~

f is constant. An

example

of such _an

expected systematic

trend is

provided

in

figure

I for the transitions

Mg+ (nS

~ 4P and Sr+

(nS

~ 5P

).

The _mean lifetime of _a

given (nl)

level that

decays spontaneously

is defined

by

I Tn< ~

£

A

(nl

~ n'l'

)

n'l'

where the sommation holds [or all states

in'l') (E~,,,

_<

E~j)

that _are

radiatively

connected to the level

(nl).

It is known that the _mean lifetimes of

high-n-levels

of

quasihydrogenic

atoms are well

represented by

the

formula[11] T~j=To(n*)~.

Tablell shows the

To and _a

parameters

we obtained from _a numerical fit of _our results

(10

_{w n} w 30

).

All of the

a values _are close to

3,

which is the value

expected

in the limit of _n

large

from consideration

on the

oscillator-strength

sum-rules

[12].

5. Conclusion.

Primarily

radial matrix elements have been

computed

for

highly-excited alkaline-earth, singly-ionised

atoms. Transition

probabilities,

oscillator

strengths

and lifetimes between _a wide range of orbital I and

principal

_n

quantum

numbers have been. evaluated.. Our modest

objective

_was -to

provide

values of these

quantities

in nr~er to assist

experimentalists

in

lK° RYDBERG STATES OF IONIZED ALKALINE EARTHS 9

estimating

the lifetimes of excited-states and the

importance

of relevant processes. It is _our

hope

that the present

predictions

will stimulate further accurate

experimental investigations, especially

for

large

atomic numbers.

References

[1] BATES D. R. and DAMAGAARD A., Philos. Trans. R. Soc. London, Sect. A 242

(1949)

101.

[2] LINDGARD A, and NIELSEN S. E., Atom. Data Nucl. Data Tables 19

(1977)

533.

[3] ^THEODOSIOU C. E.,

Phys.

Rev. A 30

(1984)

2881;

GOUNAND F., J. Phys. France 40 (1979) 457 _;

LI B. _et al., J. Phys. B: Atom. Mol. Phys. 21

(1988)

2205.

[4] KOSTELECKEY V. A. and NIETO M. M.,

Phys.

Rev. A 32

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