The correlation between optical absorption spectra and g-shifts

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The correlation between optical absorption spectra and g-shifts

M. Moreno

To cite this version:

M. Moreno. The correlation between optical absorption spectra and g-shifts. Journal de Physique,

1978, 39 (1), pp.111-114. �10.1051/jphys:01978003901011100�. �jpa-00208733�

THE CORRELATION BETWEEN OPTICAL ABSORPTION SPECTRA AND g-SHIFTS

M. MORENO

Departamento

^{de Fisica}

Fundamental,

^Facultad de

Ciencias,

Universidad de

Santander, Santander, Spain

(Reçu

^{le 21}

juillet 1977, accepté

^{le 1 S}

septembre 1977)

Résumé. ²⁰¹⁴ On

analyse

l’hypothèse de Kleeman et Farge

(J.

Physique ³⁶ (1975)

1293),

^{selon laquelle}

les valeurs de _g-g0 doivent correspondre ^aux transitions à zéro-phonon, ^{en tenant} compte ^{de la}

nature vibronique des niveaux. On montre que ^cette hypothèse ^{n’est pas correcte dans le cas d’un} couplage ^{fort et} que la

position

^des transitions

effectives,

qui ^rendent compte des valeurs de _g-g0, est proche du maximum de la bande

d’absorption.

Pour K2CuF4, le coefficient de

couplage spin-orbite

^{réduit est} estimé à 541

cm-1,

^{en accord avec}

l’échelle

d’électronégativités

^{et la série} néphélauxétique.

Abstract. ²⁰¹⁴ Kleeman and Farge’s assumption

(J. Physique

³⁶ (1975)

1293),

^which implies ^that g-shifts ^must be correlated with zero-phonon ^{lines is} discussed, considering the vibronic nature of the levels. It is shown that this assumption is incorrect in strong coupling ^{cases and the} position

of the

effective

transitions, ^{which accounts for the} g-shifts, ^{lies near to} the maximum of the

absorption

band.

The reduced spin-orbit coefficient for K2CuF4 ^{is estimated to be 541 cm-1 in} agreement ^with the electronegativity scale and

nephelauxetic

^series.

Classification

Physics ^Abstracts

76.30 ^- 78.40

1. Introduction. ^- The

g-shifts

^of

magnetic

^ions

in

crystals

^are

strongly dependent

^{on the}

position

of some associated

optical absorption

transitions.

In many ^cases of octahedral or

tetragonal

^environ-

ment, these transitions

belong

^{to the}

commonly

called d-d transitions.

The correlation between

g-shifts

and

optical absorp-

tion spectra ^has

normally

been made

assuming purely sharp

electronic levels

(S.E.L.) and, therefore, neglecting

^the electron-lattice

coupling,

^which

gives

rise to more

complex

vibronic levels. The

position

of the

effective

transitions related to these S.E.L.

and which determine the

g-shifts

^was

usually

^asso-

ciated to the maximum of the

corresponding absorp-

tion band. This is the _case, for

instance,

for the well studied

Cu(H20)6+ complex [1, 2].

However,

^{in a recent} and valuable _paper on the

properties

^of

K2CuF4,

^{Kleeman and}

Farge [3]

claimed that the

position

^{of such}

effective

transitions should be associated not with the maximum of the

absorption

^{band but with the}

corresponding zero-phonon

line in view of its

purely

^electronic

origin.

^{It is clear}

that,

while both

options

^are

practi- cally equivalent

^{in cases} with’ weak electron-lattice

coupling,

great

discrepancies

appear when this cou-

pling

^{is moderate or} strong ^{as often}

happens

^when

transition metal ions with a non-filled d-shell are

involved. In

fact,

if Kleeman and

Farge’s assumption

is

adopted

^the energy of the

effective

transition is

certainly

lower than if the criterion of the maximum is

employed. Therefore,

^as valuable information on

the

covalency

of the cluster can be got ^{from the}

g-shifts [4],

^{it is a} very

important point

^{to elucidate}

the

right

way of

correlating optical absorption

^data

and

g-shifts. Taking

^{as an}

example

^{the band}

B3

of

KZCuF4 [3],

^this band has its maximum

placed

at 8 400 cm-1 while its

zero-phonon

^{line is at}

5 300

cm-’. Assuming

^{Kleeman and}

Farge’s

assump-

tion,

^{a value} of the reduced

spin-orbit

coefficient

j*

^{= 345}

^cm-1

^is deduced from the

experimental g I I -shift.

However if the criterion of the maximum is used a

certainly higher

^value

ç*

^{= 546}

^cm-1

^is

obtained, roughly implying

^{a lower}

degree

^of

covalency

in the

ground

^level

B ig

^{as well as} in the excited levels

B2g

^and

Eg,

in view of the

study

^{on the}

g-shifts

^of

D4h complexes already reported [4].

The aim of this _paper is to elucidate the present

question, treating

^the

problem

^{in a more realistic}

way which takes into account the vibronic nature of the levels. We will show that :

1)

^The

expressions

LE JOURNAL DE PHYSIQUE. ^- T. 39, ^N° 1, ^JANVIER 1978

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003901011100

112

given

^{for the}

g-shifts by

the S.E.L.

assumption

^are

essentially right. 2)

^The

energies

^{of the}

effective

transitions,

appearing

in theses

expressions,

^{lie near}

to the maximum of the

absorption

^{band in} strong

coupling

situations.

Thus,

this second

point

invalidates Kleeman and

Farge’s assumption

^{in such}

important

cases.

2. Calculations. ^- First of all we want to

point

^out

the

auxiliary hypothesis

and the limit of

validity

of our calculations :

i)

^We will consider the electronic system ^{to be}

linearly coupled

^{to a}

single

vibrational

mode,

^whose

frequency

^{is co. For}

simplicity

^the system, ^whose

ground

^{state is an} electronic orbital

singlet,

^{will be}

considered at 0 K. The

general properties

^{of this}

model are well

summarized,

^for

instance, by

^Fit-

chen

[5].

ii) Only

second order contributions

(in energy)

to the

g-shift

will be considered.

Higher

^{order terms}

can be

easily,

^{but more}

tediously,

calculated in a

similar _way

[4].

The calculations will be

especially applied

^{to a} strong electron-lattice

coupling

^case,

which,

^as

pointed

out, presents ^a

major

^interest.

iii)

^{It will be}

supposed

that the introduction of a

magnetic

field does not alter the

configuration

^coordi-

nate scheme

already

^existent

(obviously

except ^the

splittings produced by

^the

field).

iiii) Finally,

ⁱⁿ

evaluating

^matrix elements of

purely

electronic operators, the Condon

approxi-

mation will be assumed.

Now it has to be remarked

that,

^{in order to have}

a clear

understanding

^{of the}

symbols,

^Fitchen’s

notation

[5]

^{will be}

preferably

^used.

Considering

^this

fact and

i),

the vibronic levels

tp an’ tp bm

^{of our} system will be written as :

Whose

energies

^{are :}

In these

expressions

the index « a » refers to the electronic

ground

^state while the index « b » refers to an excited one. The index n and m denote vibra- tional quantum numbers of

excitation; Qb

^{is the}

equilibrium position

^{for the}

configuration

^coordinate

in the excited electronic level « b ».

(Q2

^{has been}

chosen to be

zero.)

In the

present

discussion

lfJ a(r, Q)

^and

Ob(r, Q)

will be adiabatic electronic functions obtained consi-

dering only

the influence of the electrostatic and

crystal

field

interactions,

^but

neglecting

the influence of the

spin-orbit coupling,

which will be further treated

as a

perturbation.

^This

approach

^is

adequate

^for

transition-metal ions with

incomplete d-shells,

^but

incorrect for rare-earth ones.

Therefore,

^{at 0}

K,

^the system ^{will be}

placed

ⁱⁿ

its

ground

^level

If now a

magnetic

^{field is}

applied

^{and we consider}

the combined effect due to the Zeeman and

spin-

orbit terms, the contribution ^to the

g-shift, taking

into account

ii)

^and

iii),

^{will come from terms like}

Here

Ô1

^and

Ô2

^{denote two} operators

(such

^{as the}

Zeeman and

spin-orbit ones), which,

^{in a first} _step,

can be considred as

purely

electronic.

Now

taking

^{into account}

iiii)

^the

possible

^terms

of the summation in

(4)

^{with b = a but m # 0 are}

excluded due to the

orthogonality

^relation

Therefore

(4)

^can be rewritten as

where we have written

pu(r, 0), qJb(r, 0)

ⁱⁿ the electronic matrix elements in order ^to

emphasize

^their

indepen-

dence from

Q-coordinates.

The

preceding expression (6) clearly

shows that in the present ^{more realistic}

scheme,

^the contribution to the

g-shift

^is

quite

^{similar to the one deduced}

within the S.E.L.

approximation [1].

^{However in}

the present model the effective _energy

(Eba)eff

^is

unequivocally

^defined

by :

In the limit case of zero electron-lattice

coupling, Q b 0

^{= 0 and}

therefore, considering (5)

^and

(7), (Eba)eff

^{will be}

simply equal

^to

Ebo - E,,o (simply

called

Eo by

^Fitchen

[5]),

^which

^gives

^the

^position

of the

zero-phonon line, which, obviously,

^{will be}

the

only

^one

appearing

^{in the}

absorption

spectrum in this extreme situation.

In view of well-known results

[5], (7)

^can be rewritten

as :

where,

^{at 0}

K, Wom

⁼

^e-’ S"’/m !

^{is a} distribution

function, peaked

^{at m* = S in the} strong

coupling

case, S

being

^the

Huang-Rhys

factor. Therefore in an

opposite

strong

coupling

^{situation the}

major

contribution to

(Ebaeff

will arise from

energies

^near

to

Eba

⁼

Ebo - Eao

⁺

Shco, which,

^{as is}

known, gives

^the

position

^{of the maximum of the}

expérimental absorption

^band.

In order to

give

^{a more useful}

expression

^for

(Eba)eff (8)

^{can be}

expressed

^{as :}

Now,

^{if we take into account}

that, usually Eba > (m - m*)

1iro for all the values of m which

give

^{a non}

negligeable

contribution to

(Eba)eff, (9)

can be

approximated by

and

remembering

^that

[5]

and

putting

^{m* = S we}

finally arrive,

^{in a} strong

coupling situation,

^{at :}

M2

⁼

S(nro)2 being

^{the second moment of the}

absorption

^band.

Finally

^it should be also noticed that the

present argument

^{is not}

only

valid for the

g-shift problem ;

but it

might

be used in other ones

(such

^{as the}

hyper-

fine

interaction,

^for

instance)

where the operators involved can be considered as

purely electronic,

^at

least in a first

approximation.

3.

Applications

^and discussion. ^- We will first

apply

the

preceding

^{results to the}

B3

^band

[3]

^of

K2CuF4,

which is

correlated,

within the second order

approxi- mation,

^{to the} g

II-shift.

Considering

^the

experimental

^data

[3],

^{a value}

S1ïro ⁼ 3 100 is obtained for the

B3

band from the difference between the

position

of the maximum of the band

Eba(B3)

^z

Jo

^{= 8 400}

^cm-’

^{and that}

of the associated

zero-phonon

^line

placed

^at

5 300

cm-1.

Now

taking hw) - 195 cm-1,

^after

averaging

the values of 11m derived from the

analysis

of the vibronic structure and from the variation of the oscillator

strength

^{with the}

temperature

^shown

by

^the

B2

^band

[3],

^a value S -5 16 is now deduced for the

B3

^{band. From these}

figures

^a small value

M2 (Eba)2

^{= 8.6 x}

^10-3

^is

^found,

^{which means that}

[Eb.(B3)eff

^{= 8 328}

^cm-’

^is

^only

^{a 0.86}

^%

^lower

than the maximum of the

absorption

^{band. This}

fact

obviously justifies

^{in the}

present

^{case, the}

develop-

ment made in

(10).

Now

taking

^{into account the}

gll-shift ^expression

given by

the S.E.L.

approximation [1]

^{we will write}

the g 11 - go

value,

within the present

model,

^for

K2CuF4

^{as :.}

ç* being

the reduced

spin-orbit

coefficient whose

microscopic origin

^and

validity

has been discussed elsewhere

[4].

Now, putting

ⁱⁿ

(13)

^the

experimental

a value of

ç*

^{= 541}

^cm-1

is deduced. This

value, though

smaller than the free ion one

(ç

^{= 830}

cm-’)

is

certainly higher

^{than the} calculated

by

^Kleeman

and

Farge ç*

^{= 345}

cm-1,

^which moreover seems to be in contradiction with the

electronegativity

scale and the

Jorgensen’s nephelauxetic

^series

[7].

In fact in a similar Jahn-Teller

system CU2+ : CdCl2,

with clorine ions

acting

^as

ligands,

^{a value of}

ç*

^{= 405}

^cm-’

is deduced

[4].

^In view of the

greater electronegativity

of the fluorine ion a lower cova-

lency

^- and therefore ^a smaller reduction effect ^-

114

can be

expected

^for

K2CuF 4’

^{Our estimation}

ç*

^{= 541}

^cm-1

^{is in}

agreement

^{with this}

fact,

whereas it does not

happen

^{so with the low value}

j*

^{= 345}

^cm-1

^estimated

by

Kleeman and

Farge.

Another feature which

supports

^the

preceding

argument ^{is the}

beginning

^{of the}

experimental charge-

transfer transitions in

K2CuF4

^and

^{CU2+ :} CdCl2.

While,

in the first ^case

[3],

^this

happens

^around

38 000

cm -1,

^{in the second case} the maximum of the first

charge-transfer

^{band is}

placed

^at

a fact

which,

^as

emphasized by Jorgensen [7],

^{is also}

related to the

higher electronegativity

^of the fluorine ion.

The

understanding

^{of the} gl factor in

K2CuF4

is not so

simple

^{due to the}

following

^{reasons :}

i)

^The

more

important

^role

played by

the third order contri- bution in the

interpretation

^{of the} gl value

[4] ; ii)

The existence of a

slight

orthorhombic distortion in the

K2 CuF4

lattice which

splits

^the

Eg ^{level, giving}

rise to the

B,

^{and the}

B2 ^bands,

^observed

by

^Klee-

man and

Farge [3],

whose maximum are

placed

^at

12 200

cm-1

and 9 400

cm-1

¹

respectively.

However in order to get ^an estimation of _gl for

K2CuF4

^we will write _gl as :

where

A 1

^{is the} energy of the

B*1g

^~

^Eg transition,

which will be

simply

^taken

averaging

^the

energies corresponding

^to the maximum of the

B,

^and

B2 ^bands,

and

G 1.

^means the third order contribution to gl.

Therefore, putting 41

1 ⁼ 10 800

cm-’

and

a value 2

j*/4 1

^{= 0.100 is} obtained. As

emphasized

in

[4]

the value of

G 1.

^{in a more realistic Molecular}

Orbital

(M.O.)

^{scheme is}

higher

than that calculated

by

^{means of the}

crystal

^field

approximation.

^Therefore

we can say

implying Gl >

^{0.017 for}

K2CuF4

^which

finally

means that _gl = 2.085 is an upper estimated value for _gl, close to the

experimental

one 9-L ⁼ 2.08

[6].

This fact

points

^{out to us}

^that, although

^{this agree-}

ment may be a little fortuitous in view of the

approxi-

mations

made,

however the

general

features of the g-tensor ^of

K2CuF4

^{can be more}

adequately

^under-

stood within the

present

^model.

Nevertheless further refinements ^-

including

^a

careful treatment within a M.O.

picture

^{- in order}

to

get

^{a more} strong correlation between

E.P.R., N.M.R.,

^neutron diffraction and

optical

^data

[3, 6, 9],

are needed. An account of this

work, which,

^{at the}

present,

^{is in} progress, will be

reported

^{in the near}

future.

References

[1] ABRAGAM, A., BLEANEY, B., ^Electron Paramagnetic ^Resonance of transition ions (Clarendon Press, Oxford) 1970, p. 401 and 456.

[2] MORENO, M., Solid State Commun. 18 (1976) ^1067.

[3] KLEEMAN, W., FARGE, Y., ^J. Physique ³⁶ (1975) ^1293.

[4] MORENO, M., J. Phys. ^{C 9} (1976) ^3277.

[5] FITCHEN, D. B., in Physics of Color Centers, ^Ed. by ^{W. B. Fowler} (Academic Press, ^New York) ^1968.

[6] YAMADA, I., ^J. Phys. ^Soc. Jpn ³³ (1972) ^979.

[7] JØRGENSEN, C. K., Solid State Phys. ^{vol. 13} (Academic Press, New York) 1962, p. 425 and 431.

[8] KAN’NO, K., NAOE, S., MUKAI, S., NAKAI, Y., ^{Solid State} Commun. 13 (1973) ^1325.

[9] TOFIELD, ^B. C., ^J. Physique Colloq. ³⁷ (1976) ^{C 6-539 and}

References therein.