Hartree-Fock evaluations of specific-mass isotope shifts

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Hartree-Fock evaluations of specific-mass isotope shifts

J. Bauche

To cite this version:

J. Bauche. Hartree-Fock evaluations of specific-mass isotope shifts. Journal de Physique, 1974, 35

(1), pp.19-26. �10.1051/jphys:0197400350101900�. �jpa-00208122�

HARTREE-FOCK EVALUATIONS

OF SPECIFIC-MASS ISOTOPE SHIFTS

J. BAUCHE

Laboratoire Aimé

Cotton,

^CNRS

II,

^Bâtiment

505,

⁹¹⁴⁰⁵

Orsay,

^France

(Reçu

^{le 23}

juillet 1973)

Résumé. ^- A la suite d’un travail

précédent

^{sur la série} 3d, ^des

déplacements spécifiques

de

masse sont

calculés,

par la méthode de Hartree-Fock non

relativiste,

^{dans des atomes des séries}

4d, 4f,

^{5d et}

5f,

pour les raies de résonance des alcalins et dans

quelques

spectres ^{à deux et trois} électrons. Dans les atomes lourds,

quand

^on

dispose

de résultats

expérimentaux

^{sur les} spectres ^X d’atomes

muoniques,

^on compare directement

expérience

^et résultats de Hartree-Fock _pour les raies

optiques

^à

grand déplacement spécifique.

^{Dans les autres} cas, ^en utilisant des

diagrammes

de

King,

^on obtient les valeurs

expérimentales

de certaines combinaisons linéaires de

déplacements spécifiques

^{de deux}

raies,

que l’on _compare aux résultats de Hartree-Fock.

Dans toutes les séries

longues nlN,

^valeurs

expérimentales

^et

théoriques

^{diffèrent au}

plus

^d’un

facteur deux. Le résultat essentiel de la

théorie,

^{à savoir la}

prédiction

que les transitions corres-

pondant

^{au saut} d’un électron nl ont de loin les

plus grands déplacements

^de masse, est ainsi confirmé.

Abstract. ^-

Following

^a

previous

paper ^on the 3d

series, specific-mass isotope

^{shifts are} comput- ed, by ^{means of the} non-relativistic Hartree-Fock

method,

^{in atoms of the} 4d, 4f, ^{5d and 5f}

series,

in the alkalis and in a few two- and three-electron spectra. ^In

heavy

atoms, ^when

isotopic

^measure-

ments of muonic X-ray transition lines are

available,

^direct

comparisons

^with

experiment

^are

presented

^for

optical

transitions with

large specific

shifts. In the other _cases,

through

^{the use of}

King diagrams,

^the

comparison

is made between the Hartree-Fock

(HF)

^and

experimental

^values

of some known linear combination of the

specific

^{shifts of two lines.}

In all nlN

long

series, ^{the HF and}

experimental

^values agree within a factor of two. This

gives

confidence in the main HF result, ^{i. e. the}

prediction

that the lines with the

jump

^{of an} nl electron have by ^{far the}

largest

^{mass shifts.}

Classification

Physics ^Abstracts

5.231 1

1. Introduction. ^- Measurements ^on the

isotope

shifts of samarium

by Striganov,

Katulin and Eliseev

[1 ]

and their

interpretation by King [2]

^{renewed ten}

years ago the interest in the atomic mass

isotope

shift.

Through

these results it ^was known that

specific

mass

isotope

^{shift may be of}

appreciable

^influence

even in the

spectra

^{of atoms with} atomic number Z

larger

^{than 60.}

Consequently,

^for

people

^interested

in field

isotope shift,

^{the ab initio} evaluation of the

specific

^mass

isotope

shift is of

importance,

^because

its value ^can then be subtracted from the

experimental

result to isolate the field shift contribution.

Except

^for very

light

^atoms,

specific

^{mass shifts}

have

generally

^{been evaluated}

through

^{the use of}

the Hartree-Fock

(HF)

method. In

1970,

^{Bauche and} Crubellier

[3]

^have

published

extensive results obtained in that _way in the arc

spectra

of the first series of transition metals

(the

^3d

series).

^{In this}

series,

^the

field shift is of

relatively

^small

importance

^{and can}

be considered as a correction. So the

comparisons

between

experiment

^and

theory

^{for the}

specific

^shift

are safe when the

isotope

^{shifts are}

large.

^{For the}

transitions with the

largest shifts,

it appears that

experimental

and theoretical values of the

specific

shifts _agree to about 30

%.

Encouraged by

^this

result,

^{we have}

proceeded

^to

elements with

larger

^values of Z. This paper

presents, first,

results for the first series of rare earths

(the

^4f

series).

Then the other three

long

^series

(4d,

^{5d and}

5f)

are

studied,

each series

through

^the

example

^{of one}

element.

Eventually

^we

present

evaluations for the first resonance lines of the alkalis and for miscella-

neous one-, two- and three-electron

spectra.

^Wherever

possible, comparisons

^between

experimental

^{and theo-}

retical results are

given. Unfortunately,

^{it is not}

known in which _way the HF results are affected

by

the

relativity phenomenon.

2.

Principles.

^{- We} first recall the

general

proce- dure

applied.

^{The total}

specific

^{shift 6 of a state}

is,

to first order of

perturbation,

^the

expectation

^value of the

specific

^shift

operator

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197400350101900

20

where pi is the momentum of electron i and M the

mass of the nucleus.

Knowing

^{the values}

^{of ù}

^for

levels A and B and nuclear masses

Mi

^and

M2 (M2

>

MI),

^the

specific

^{shift DQ} for transition B ---> A

(with

the usual

sign conventions)

^is

where

ul,(A)

⁼

6(A, M2) - u(A, Ml)

^{is the}

specific

shift for level A.

From now on we deal with atomic wavefunctions of the

configurational angular type (e.

^g. the Hartree- Fock

(HF) wavefunctions).

^A

^previous

^paper

^[3]

recalls the _way to build the formal

expression

^{of r}

in terms of the radial

integrals

For a state in pure

LS coupling,

^this

formal

expres- sion does not

depend

^{on the}

quantum

^{numbers J} and

MJ

^{of the state}

[4]. Furthermore,

^{it does not}

depend

^{on its}

intermediate-coupling expansion

^{in the}

(pure) configurations

^{where no}

^G’

^Slater

integral

^is

present

^{in the formal}

expressions

for the relative electrostatic

energies

^{of the terms}

[5]. Examples

^of

this last case are

dN s2, ^{dN+ 1}

s,

fN S2, pN and,

^of course, the so-called one-electron

configurations.

For any

monoconfigurational

^{atomic LS}

state,

^the HF method

yields

radial functions

R(n, 1)

^{for the}

various subshells. From these

functions,

^{it is}

straight-

forward to

compute

^{the J}

integrals

^{and then -u. We}

have used the very convenient HF

computer

^{code of} Froese

[6],

^which

yields

the radial functions in nume-

rical form.

Apart

^from

simple

^numerical

data,

^the

required input

^consists in the formal

expansion

^of

the electrostatic _energy, in terms of Slater

integrals,

for the considered atomic LS term.

3. HF results. ^- All the results

presented

^hereafter

refer to atomic _{pure LS} terms.

Tables

I,

^II and III contain

respectively

the results obtained in the arc

spectra

^{of the 4f}

series,

^of

molyb-

denum

(4d series),

^osmium

(5d)

^and

plutonium (5f)

and of the alkalis and miscellaneous one-, two- and three-electron

spectra.

^{In each}

table,

^{the first three} columns indicate

respectively

^the

elements,

^{the chosen}

pairs

^of

isotopes (mass

^numbers

A1

^and

A2,

^with

AA

= A 2 - A 1 = 2)

^{and for} each element the LS terms

considered ;

the last three columns contain

respectively,

^{for each LS} term, ^the

computed

^value

of the

quantity k

^{= -}

(M/m) u

^{in atomic energy units}

(notation

from Vinti

[4] ;

^{m is the} electronic _{mass ;} k is ^a

purely

electronic

quantity,

^not

depending

^on

nuclear

masses),

the shifted value of k when that of the first LS ^term considered for each element is

arbitrarily

^{chosen to be} zero, and

eventually

^the

specific

^shift

Q12

^{in mK}

(1

^{mK =}

^10-3 cm-l)

^{for the}

chosen

pair

^of

isotopes (it

^can be shown that this last

figure

^{is the}

product

^{of the}

preceding

^one

by

^the

number 239

078/A1 A2).

In each

spectrum,

^we have selected for

computation

low

configurations

^{which are}

spectroscopically impor-

tant. In

configurations

^{with more than one LS} term,

we have

generally

^{selected for}

computation

^{at least}

the lowest term, ^{or at least a term} with the maximum

spin.

For

deciding

^{whether the}

specific

^{shift in a line is}

«

large »

^{or « small} », its value is often

compared

^to

the normal-mass shift value. In the tables

given,

^this

comparison

^{is most}

easily

^obtained

by noting

^that

the normal mass shift in _any line at ⁼ 5 000

Á

is

equivalent

^{to a} contribution close to 0.1 to the Ak

quantity

of the line

(Ak being

the difference between the k values of the levels

involved).

3.1 THE 4f SERIES

(TABLE 1).

^{- In the results for}

the 4f

series,

^a

major phenomenon

appears : in each

spectrum,

the value of the

specific

^shift

depends essentially

^on the number of 4f electrons

present

ⁱⁿ the

configuration.

^This

phenomenon

^is

quite analogous

to that observed on the results for the 3d series

[3].

The

larger

the number of 4f

electrons,

^the

larger

^the

specific

shift of the term ; moreover, the increase in

specific

shift per 4f electron has almost the ^same value

(30 mK,

^{for AA =}

2)

ⁱⁿ any ^arc

spectrum

^of the series. In contrast, ^{if two}

configurations

^differ

only

in the number of

6s, 6p and/or

^5d

electrons,

their

specific

^shifts

only

^differ

by

^{a few}

mK,

^{at most.}

3.2 THE

4d,

^{5d AND 5f SERIES}

(TABLE II).

^{- For}

each of the

long periods

other than 3d and

4f,

^we

limit ourselves to the

study

^{of one element near the}

middle of the series.

From the results of table II it _appears at once that in these three

nlN

series

again

the Hartree-Fock value of the

specific

^{shift of a}

configuration depends

^almost

only

^{on its} number of nl electrons.

3. 3 THE ^ALKALIS. MISCELLANEOUS SIMPLE SPECTRA

(TABLE III).

^{- In the}

^alkalis,

^we

give only

results for the first resonance lines _np ---> ns. As from sodium to cesium there is ^a

large

variation in atomic

weight,

it is better to look at the values of Ak

(fifth column)

than of 4W

(sixth column).

^{Then it is} clear that the

predicted specific

^{shift is not}

larges 1 Ak is

^{less than}

0.1,

^as it is in the lowest np ^{--+ ns} transitions of the

long

series of tables 1 and II.

In Table

III,

^{results on some} miscellaneous confi-

gurations

^{are also}

presented.

3.4 ^DETAILED CONTRIBUTIONS OF THE DIFFERENT SUBSHELLS

(TABLE IV).

^{- To}

give

^an idea of the _way in which the different subshells of the

configurations

contribute to the k values of tables 1 to

III,

^we

present

in table

IV,

^{for a few}

typical

cases, the

separate

contributions for the interactions inside the core

(denoted core-core),

between the core and the different

external subshells

SI, S2,

^S3

(core-S1, S2, S3)

^and

between the external subshells

(external-external).

On this

table,

^{one can} observe for

example,

ⁱⁿ any

nlN

séries : ^,

- the

major

^{influence of the nl}

electrons, ,

- the

screening

^{effects on the external} non-nl elec-

trons,

- the relaxation of the core and the

self-screening

effect in the

nlN subshell,

^{which are essential}

pheno-

mena for

quenching partially

the effect of the nl-elec- tron

jump.

TABLE 1

TABLE II

22

TABLE III

TABLE IV

4.

Description

^{of the}

comparisons

^with

experi-

ment. ^- 4.1 PRINCIPLES. ^- In the _very

light

^ele-

ments

(Z 20),

^the

experimental isotope

^{shifts are}

practically

pure ^mass shifts. In the elements of the 3d series

[3],

^{the field}

isotope

shift is much smaller than the

large

^mass shifts due to the

jump

^{of a 3d elec-}

tron ; therefore it can be subtracted in an

approximate

way ^to isolate the mass effect with ^a

good

accuracy.

But now, in the

4d, 4f,

^{5d and 5f}

series,

^{the field}

isotope

^{shift is}

certainly

^{the dominant} contribution

to the

experimental isotope

^{shift in most}

optical lines.

So we consider

that,

^{in the}

medium-weight

^and

heavy ^elements,

^the

only

^safe way to extract ^some information on the

specific

shift from the

experimen-

tal results is the use of

King diagrams.

^{We refer to}

the paper

by King [7]

^{for the}

practical

way to ^use this well-known method.

4.1.1

King diagrams

^with

optical lines.,- Firstly,

we are interested in

King diagrams

built with results obtained

solely

^on

optical

lines. In these _cases, each

King

^line built from the measurements on

optical

lines a and b

yields

the numerical values of the linear combination

(notations

^{as in}

^{[8] ;} (0

^{is the}

intercept

^{of the}

line),

where

Ka (resp. Kb)

^{is the}

product by Al A2/(A2 - A1) (with A 2 > A 1)

^{of the mass} shift between

isotopes

¹

and 2 in line a

(resp. b)

^{and where} p is the

slope

^{of the}

King

line. In the

following,

^{we use a}

quantity

^which

we denote

Sp(a, b),

deduced from

(o(a, b) by

^subtract-

ing

^the

easily-calculated

contribution of the Bohr

mass

isotope

shift. In terms of the

specific

^shifts

^Au

in the

lines,

^{we have ’}

To compare

experiment

^and

theory,

^{we choose to}

compare

experimental

values of

Sp(a, b) (obtained

from

King lines)

^{with the}

corresponding

^{HF values}

(which, actually, depend

^{on the}

slope p

^{of the}

King line,

^an

experimental quantity).

4.1.2

King diagrams

^with

optical

^{and muonic lines.}

-

Secondly,

we can use

King diagrams

^{built with}

results obtained on both

optical

and muonic

X-ray

lines. A

King

line of this

type yields practically

^a

numerical value of the mass shift

Ka

^{in the}

optical

line

[8].

^{In the}

following,

this allows direct

compari-

sons between theoretical and

experimental specific

shifts in Nd and Mo.

However,

^{due to} the difficulties in the evaluation of the

nuclear-polarization

^correc-

tion to the muonic shifts and in view of other pro- blems

[9],

^the

question

^{as to} whether the relative

isotope

shifts in field

optical

and muonic lines are

close

enough

^{is not}

yet

^settled.

4.1.3

Close-configuration mixing effects.

^{- The}

formula

given

^{above for}

Sp(a, b)

^{leads to a}

straight-

forward HF evaluation in the cases where the a and b lines occur between pure LS ^terms of _pure

configu-

rations.

But,

^{in the}

complex spectra, close-configu-

ration

mixing

^and intermediate

coupling

^are

impor-

tant

phenomena.

^{Both can} be studied at the same

time

through

^a

parametric study,

when the number of measured levels is sufficient. It has been observed

on a few cases

[10]

^that

configuration mixing

^has

by

far the

strongest

^effects.

Therefore,

^{for the}

experimen-

tal

comparison,

^{we tend to} select lines

corresponding

to levels for which the

configurational

identification is

supported theoretically by

^some

configuration- mixing study.

For a level whose wavefunction is known to be

a Ycl + PtJ’C2 (tpCi

^{is an}

intermediate-coupling

^wave-

function inside

configuration CJ ; ^a2

⁺

p2

⁼

1),

^we

consider that ^a

satisfactory approximation

^{to the}

theoretical value of the

quantity k

^is

This is an

application

^{of the}

« sharing

^{rule »}

[10]

which has been known for a

long

^time

[11].

To

conclude,

^{the HF} values of the

specific

^shifts

of the levels and of

(,)(a, b)

^are

computed,

^{in the}

following,

^{from the} results in tables 1 and

II,

^the

intermediate-coupling expansions

^{of the relevant}

levels

(when available)

^{and the}

slope p

^{of the}

King-line (when necessary).

^The

intermediate-coupling effects,

linked to the variation of the

specific

shift from term to term in a

given configuration,

^{can be}

neglected

ⁱⁿ

a first

approximation (see 5.1).

4.2 THE 4f SERIES. ^- For the 4f

series,

^all compa- risons between

theory

^and

experiment

^are

gathered

in table V. Lines « a » of this table

correspond roughly

TABLE V

24

to the

jump

^{of a} 4f electron and

optical

lines « b » to the

jump

^{of a}

6p

^electron.

4.2.1 Cerium. ^- For the case of

cerium,

^{we refer}

to a very ^recent paper

by Champeau [12].

^{In this}

paper, ^a

King

line is built from transitions

in Ce 1

(line a)

^and

^4f2 6p -> ^4f2

6s in Ce II

(line b).

The HF result obtained

by Champeau

^for

^Au

^{in the}

first transition

(38.3 mK)

^{is almost}

equal

^{to the value}

37 mK which we deduce from table 1 for the transi- tion

4f2

5d 6s

5K --->

4f 5d

6s2 3H.

His final compa- rison between

theory

^and

experiment

for the value of

Sp(a, b)

^is

reported

^{in table V on the}

present

paper.

4.2.2 Samarium. ^- For the case of

samarium, Hansen,

^{Steudel and Walther}

[13]

have drawn

King

lines with transitions 03BB ⁼ 5 088

Á (line a)

^{and 5 252}

and 5 271

Á (two examples

^{of line}

b),

all of the

type

(4f6

^6s

6p

⁺

^4f5

^5d

6s2)

^-+

^{4f6 6s2 7F.}

^{For our} pur- pose, ^we

prefer

^to

exchange

^rôles

of 03BE and (

^{in their}

graph,

^{i. e. to take the}

graph symmetrical

^{of their}

graph

^with

respect

^to the first

diagonal.

^{The wave-}

function

expansions

of the upper

levels, resulting essentially

^{from the}

mixing

^of

4f6(7F)

^6s

6p

^with

4f5

5d

6s2,

^{are known} from the work

by Carlier,

Blaise and

Schweighofer [14].

4.2.3

Dysprosium.

^{- Dekker et al.}

[15]

^measured

isotope

shifts in the

Dy

¹

spectrum,

^{on six différent}

lines,

^{between seven}

isotopes

^{and with an} accuracy around 0.5 mK. Five of these lines ^are suitable for our

study : they

^{link the}

ground

level of the

Dy

¹

spectrum, 4fi° 5s2 5I8 (for

^which

close-configuration mixing

^is

certainly small),

with three levels

belonging

^to

4f 9(6H

⁺

6F)

^5d

^6s2

for about 96

%

^{and two other}

to

4f10(5I)

^6s

6p

for about 98

%.

These identifications have been obtained

recently by Wyart [16] through

a

parametric study

^{of the}

mixing

^{of the two}

quoted subconfigurations.

^{We choose}

arbitrarily

^{the line}

03BB ⁼ 5 652

Á (line a)

among the first three lines and 03BB ⁼ 5 547

Á (line b)

among the last two, ^because the other lines

bring

^almost

exactly

^{the same infor-}

mation.

4.2.4

Neodymium. -

Bruch et al.

[8] published

a

King diagram

^for

neodymium

^with

optical

^and

muonic

X-ray

^lines.

Among

^{the two}

optical lines, only

the line 03BB, ⁼ 5 621

À

was identified

by Wyart [16],

who ascribed it to the

[4f3(4I) ,

^5d

6S2] ’H 3 ---> 4f4 ^6s2 5 14

transition. The

experimental

value for the mass shift is - 23.8

mK,

deduced from

figure

^{1 of reference}

[8].

Because the

mixing

^of

^4f3

^5d

^6s2

^with

^4f4

^6s

6p

^is

unknown,

^this

figure

^{is an}

algebraic

upper bound ^to the true value of the mass shift for the

transition.

4. 3 THE

4d,

^{5d AND 5f SERIES. -} 4. 3. 1

Molybde-

num. ^- In the

spectrum

^{of Mo}

I,

^recent results have

been obtained

by Aufmuth, Wilde,

^{Behrens and}

Clieves

[17].

Combined in a

King diagram

^{with the}

measurements on muonic

X-ray

^lines

by Macagno

and Chasman and coworkers

[18], they yield

the spe- cific shifts of the transitions between

septets

^of

4d5 5s, 4d5 5p

^and

^4d4

^5s

5p

and between

quintets

^of

^4d5 5s, 4d5 5p

^and

^4d4 Ss2.

For the shifts of all four

configu-

rations to be linked

together,

we can assume that the

experimental

difference between the

specific

^{shifts of}

4d5

5s

7S,

^and

’S2

^{for the}

isotopic pair

96-98 is close to the HF value

(see

^table

II).

Indeed those levels are

probably

rather pure, because ^no other

7S

^or

’S

term exists in the

perturbing

^even

configurations ^{4d4 5s2}

and

4d6.

For the odd terms

4d5 5p’P

^and

^4d4

^5s

5p7p,

there remains the

problem

^{of their}

mixing through

^the

electrostatic-interaction

operator

^G.

Using

^{the for-}

mal

expression

^{of the}

off-diagonal

element of the

operator G,

^{the values of}

R’(4d 5p,

^5s

5p)

^and

R’(4d 5p, 5p 5s)

^{in the}

neighbouring

^{Tc 1}

spectrum

and the

experimental energies

^{of the terms, we find}

that the

experimental

^term

^4d5 Sp’P

contains about 7

%

^{of the term}

^4d4

^5s

5p 7 P.

This allows us to deduce from the measurements the «

experimental » specific isotope

^{shifts of the} pure terms,

by

^{use of the}

sharing

rule.

Eventually

^we

present figure

^{1 for the}

comparison

between

theory

^and

experiment.

FIG. 1. ^- Comparison ^between experiment ^and theory ^{in Mo 1} (values ⁱⁿ mK).

4.3.2 Osmium. ^- In Os

1, optical configurations

built on

5d5, ^5d6

^and

^{5d 7}

^{cores appear.} Their Hartree- Fock

specific

^{shifts are}

roughly

ⁱⁿ arithmetical _pro-

gression.

^{For the}

comparison

^with

experiment,

^we

refer to a recent

study by Champeau

^{and Miladi}

[19]

on the

nearby

^element

tungsten.

These authors have measured the shifts in the lines 03BB ⁼ 4 982

A (line a ; 5d4

6s

6p -+ ^5d4 6s2)

^{and = 4 269}

^À (line b ; 5d3 6s2 6p -+ ^5d5 6s)

^for

isotopes 180, 182,

^{184 and}

186.

They

deduce from the

corresponding King-

graph

^the

experimental

^value

Sp(a, b) = - 70 + 34 K,

to be

compared

^to the Hartree-Fock value - 96 K.

4.3.3 Plutonium. ^- Plutonium is the heaviest ele- ment where mass

isotope

^{shift has ever been observed.}

The results of Tomkins and Gerstenkorn

[20]

^{for the}

lines 03BB ⁼ 5

712,4 Á (line a ; ^5f5

^6d

^{7s2 -+ 5f6} 7s2)

^and

03BB ⁼ 4

735,4 Á (line b ; 5f 6 ,7s 7p

^--+

^5f6 7s2)

^{can be}

used to build a

King graph,

^which

yields

^the

experi-

mental value

Sp(a, b) = -

^{497 ±} 120 K. The

mixing

of the relevant odd

configurations

^{has never been}

studied,

^but the values of the shifts

themselves,

^essen-

tially

^{due to field}

isotope effect,

indicate that the upper level of line b is

appreciably

mixed with

5f5

6d

7s2.

The HF value of

Sp(a, b), -

^{287 K}

(obtain-

ed from the results of table II and the

assumption that,

^as

everywhere

else in this paper, the np ^{i ns tran-} sition has a very small

specific shift),

^{is an}

algebraic

lower bound to a correct theoretical

evaluation, namely

^an evaluation

taking

^the

mixing

^{into account.}

4.3.4 Two-electron spectra. ^{- A direct}

comparison

with

experiment

^{can be}

given

^{for the}

specific

^{shift A-u}

in the transitions 3s

3p 1,3’P --> 3S2 IS

of

magnesium

and 4s

4p 1P --> 4s2 1S

of calcium

(Table VI).

TABLE VI

5. Discussion and conclusion. ^- 5.1 VARIATION OF Q INSIDE A PURE CONFIGURATION. ^- The influence of

configuration mixing

^{on the}

specific

shift has been studied above

(§ 4.1.3).

We consider now the

intermediate-coupling pheno-

menon. This

phenomenon

would have no influence if the LS terms of the considered

configuration

^had

the same

specific

shift. So the

question

is : does the HF

specific

^shift

change appreciably

^{from one LS}

term to the other in a

given configuration ?

A

part

^{of the answer is}

given

ⁱⁿ

paragraph 2,

where ^a difference is noted between the

configurations

where the formal

expression

^{of à does not}

depend

on the LS term and those where it does.

Now,

^{the HF method is not a} central-field method.

For two LS terms of the same

configuration

^with

different formal

expressions

^{of the} electrostatic energy, the two sets of HF radial functions are not

exactly

^the same ; so, ^even when the formal expres- sions of (i are

identical,

their numerical HF values

generally

differ. In the strict central-field

scheme,

these

changes

^{can be} viewed upon ^as

far-configuration mixing

effects. A numerical

example

^{can be found}

in Mo 1

4d55s (Table II) :

the theoretical

specific

shifts in the terms

’S

and

5S

differ

by

^{3 mK for}

AA ⁼

2,

which is small with

respect

^to the effect of the

jump

^{of a 4d}

^electron,

^{but not}

negligible.

As concerns

configurations

^with

^G1 integrals

^for

open-shell electrons,

^we

present

^{four terms in Nd}

I,

all in the

coupling (4f3 4I,

^5d

6s2).

^{Due to the 4f-5d}

interaction

[23],

the formal

expressions ^{of u}

^{are all}

different. But the

changes

^{in the HF} radial functions cancel almost

exactly

the effect of the formal

changes

on the numerical value of 7 : the

resulting

^numerical

specific

^{shifts are} almost identical in this

particular

case.

Eventually,

^to

get

^{an idea} of the first-order varia- tion of à inside a

configuration,

^{i. e.} the variation due to the

changes

^{in its formal}

expression,

^{we can}

look at the squares of the J

integrals,

^{which are}

listed in table VII. In the 4f

series, only

the values

TABLE VII

for samarium are

given,

because those for the other lanthanides are almost the same

(but

^for

^cerium,

where

J’(4f, 5d)

^{is much}

smaller).

We also know

that,

^{from one term to} another in a

given configuration,

the

changes

in the coefficients of these

J2 quantities

in the formal

expansions

^{of k are at most of the order}

of 1. So we conclude that the variation of the HF

specific

shift from term to term in a pure

complex configuration

^{of the}

long-period

elements is at most

of the order of the normal mass effect in ^an

optical

line of the

spectrum.

5.2 COMPARISON BETWEEN THEORY AND EXPERI- MENT. ^- Some

comparisons

^with

experiment (for ^Nd,

Os, Mg

and Ca above and in

[3])

^{have been}

^presented

directly

^on

specific-shift

values. All the others above

(rare earths)

^concern

quantities Sp(a, b). However,

one sees that there are

practically only

^two

types

^of

specific-shift

values in the HF results for the

interesting

pure

configurations

^{of each} rare-earth

spectrum :

^the difference between these shifts is characteristic of the

jump

^{of an} f electron. We deduce

that,

whatever the

(accounted for) configuration mixing,

^{the tests on}

Sp(a, b)

^{can be} looked upon ^{as tests on} the value relative to an f-electron

jump.

26

Then we come to the

following

conclusions. In _any

spectrum

^{of the}

long

^{series nl}

(1

^{= 2 or}

3),

^{for a tran-}

sition where an nl electron

jumps,

the Hartree-Fock method

yields

^{an ab initio} evaluation of the

specific

mass shift which is correct in

sign

^and

approaches

^the

experimental

value inside a factor of two. Such transi- tions are those for which the theoretical value of the

specific

^mass shift is around ten times or more the normal mass shift.

More

precisely

^{the HF} value appears ^to be close to 1.5 ^or 2 times the

experimental

^{value in the}

spectra

^of the

3d, 4f,

4d and 5d series.

Figure 1,

^{in which we}

present

^the

comparison

^for

molybdenum,

^{offers a}

striking similarity

^{with the}

analogous figures

^{in the}

1970 paper

[3]

^{for the cases} of Ni 1 and Cu I. Even-

tually,

^{in the}

example

^we

present

^for

plutonium,

^the

experimental

^{value is}

larger

than the theoretical one.

In the tables of this paper there also appear results for another

important type

of transitions : for the lowest _{np --+} ns

jumps

^of any

spectrum (except maybe

in the _very

light elements),

^the

specific

^{mass shift is}

predicted

^{to be at most}

equal

ⁱⁿ

magnitude

^{to the}

normal mass shift.

Experimental

^evidence for this is

given only

^for

magnesium

and calcium

(Table VI),

but with _poor numerical

agreement

^between

experi-

ment and

theory.

However it must be noted that all

our

comparisons

^between

experiment

^and

theory

^{in a}

long

^nl

series,

but those for

neodymium

^and

molybde-

num, deal with a

couple

^of

optical transitions,

^one

with

practically

^the

jump

^{of an nl electron and the}

other with a np -+ ns

jump

of interest in this discus- sion. These

comparisons

^{must therefore be}

considered,

in

principle,

^{as tests} for the combined assertion that the

specific

^{shift is}

large

^and

positive

^{for the}

jump

^of

an nl electron to another subshell and that it is small for the lowest _p --> s

jump.

Much work will be needed in the future to obtain accurate ab initio evaluations of the

specific

^{mass shift}

in

medium-weight

^and

heavy

^{atoms. More}

precisely,

progress is necessary in the

computation

^{of both}

correlation effects and relativistic effects. For correla-

tion,

^{a recent} paper

by

^Labarthe

[24] already

^shows

that it is not sufficient to

compute

the crossed second- order effects of 03A3

(the specific shift)

^{and G}

(the

^elec-

trostatic

interaction).

^{As concerns}

relativity,

^{no result}

is

yet

^{available on its influence on the}

specific

^{shift in}

heavy

^atoms.

Acknowledgments.

^{- We}

gratefully acknowledge

the ^use of the HF numerical

computer

program of C. Froese-Fischer for

obtaining

the radial electronic functions which ^are the basis of this work. Thanks

are also due to the

experimental

^{group at Hannover}

for communication of their

results,

^{and to R.-J.}

Champeau

and W. H.

King

^for

helpful

^sugges-

tions.

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