Radiofrequency and Mössbauer Spectra of a Paramaghetic Ion in the Presence of Hyperfine Interactions and of Tunnel Effect

(1)

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Radiofrequency and Mössbauer Spectra of a Paramaghetic Ion in the Presence of Hyperfine

Interactions and of Tunnel Effect

Françoise Hartmann-Boutron

To cite this version:

Françoise Hartmann-Boutron. Radiofrequency and Mössbauer Spectra of a Paramaghetic Ion in the

Presence of Hyperfine Interactions and of Tunnel Effect. Journal de Physique I, EDP Sciences, 1996,

6 (1), pp.137-147. �10.1051/jp1:1996133�. �jpa-00247173�

(2)

Radiofrequency and Môssbauer Spectra of

_a

Paramagnetic Ion in the Presence of Hyperfine Interactions and of Tunnel Elfect

Françoise

Hartmann-Boutron

Laboratoire de

Spectrométrie Physique(*),

Université J. Fourier Grenoble I, B-P. 87,

38402 Saint Martin d'Hères

Cedex,

France

(Received

13

July1995,

revised in final form and accepted 2 October

1995)

Abstract. The influence of _a coherent tunnel elfect _on the magnetic

hyperfine

structure of

a diluted

paramagnetic

ion, with

an electronic

ground

state

[S;

₌ +S >

(S

>

)),

is studied under the

assurnptions

of _zero

longitudinal rnagnetic

field and slow electronic relaxation. It is shown that the

radiofrequency

and Môssbauer spectra ^should be

altered,

with _a

change

in the number and intensities of the hnes. When the

tunnelling

parameter, r, ^is

large compared

to the

hyperfine

_structure, the

magnetic hyperfine

structure

sphtting

should be

strongly

reduced and

it should vanish for r _- _cc~.

Tunnelhng

elfects have already been observed in EPR expenrnents

on Tb~+ _diluted _in diamagnetic matrices. The _cases of NMR with f

=

3/2

and of Môssbauer

spectroscopy for _a transition le

=

3/2

_fg ₌

1/2

_are considered.

PACS. 76.20+q General

theory

of

resonances and relaxations.

PACS. 75.60Jp

Fine-partiale

systerns.

PACS. 73.40Gk

Tunnehng.

1. Introduction

The

tunneling properties

^oi _a

large quantum

^electronic

spin

S have been

recently

studied in reierences

il-Si-

A

large

_spin _may

correspond

to the resultant spin oi _a small ferro _or

ferrimagnetic particle,

_or to the resultant _spin oi _a well defined cluster oi 10-20 transition ions in the _core oi _a

big

_organic

molecule;

such _organic molecules _can be

prepared

in

single crystal iorm,

with _very weak interactions between different dusters

[6-8j.

The smallest

possible

such duster is _a

single paramagnetic

ion diluted in _a

diamagnetic

matrix. This _case _is

interesting

because it has been shown:

1) that the coherent

tunneling splitting

decreases

exponentially

with the number N oi atoms oi the duster

[1-4j,

and _con be

expected

to be _very small _even for _a iew _atoms

(<

Hz for

"Mn12" (4, Si),

ii)

that the

couphng

oi the giant spin with _a

large

number oi nuclear spins has _a deletenous effect _on

tunneling phenomena

_[9j

[10]

[SI. ^In a

single paramagnetic ion,

the electronic

spin

is

appreciably coupled only

with its _own nudear

spin

and these

problems

should not be

important.

(*) UJF-CNRS,

UA 08

g

Les

Éditions

_de

Physique

1996

(3)

Note that the situation considered here differs somewhat irom that oi _a small

antiierromagnetic particle,

where the

tunneling splitting

involves different mechanisms and is

distinctly larger iii il [12],

also [13] for ^the effect oi nuclear

spins).

Let _us _now retum to the

problem

oi interest here and [et _us _assume that the electronic

spin

is well described

by

_a main axial Hamiltonian:

lis

"

-D'S) il)

with D' _> 0.

(Note:

for Rare Earth

ions,

S should be

replaced by J.)

The lowest

eigenstate

oi

lis

is the doublet

[Sz

= +S _>. In _zero

longitudinal field,

and in the presence of tunnel

effect,

the

degeneracy

oi the doublet is

liited, giving

rise to a

symmetric

state

~~

and _an

antisymmetric

state

~~:

separated by

_an _energy 2r

(the "tunnelling splitting"). Expressions

of 2r

corresponding

to

crystalline perturbations

with varions

symmetries (orthorhombic, tetragonal, etc.)

_or to _a

transverse

magnetic

^field ^have ^been ^derived ⁱⁿ ^references

[il

_{[3] [4]}

_[Si.

~~

and

~~

_are

diamagnetic

states, ^but <

il~[Sz[~~ >#

0. Therefore EPR between

~~

and

~~

should be observable

by applying

_a

radiofrequency

^field

parallel

to Oz. Such

experiments

have been carried _out

recently by

Awschalom et ai.

[14]

on

antiferromagnetic ferritin,

which is

a

large

molecule with _a small

uncompensated

moment c~

@,

and for

which, apparently,

2r _m

a few MHz

([14j, Fig. 3). However, tunnelling splittings

A _e 2r _m 10 30 GHz have

already

been observed in the

past

on the much

simpler

_case of _a terbium ion

(Tb~+,

J

=

6,

1=

3/2),

diluted in _a

diamagnetic

matrix such _as

yttrium ethylsulfate _[4].

Because of the nuclear

spin I,

the EPR

signal

in _zero external field is in fact

composed

of two lines

corresponding respectively

to

Iz

₌ M

=

+3/2

and

+1/2.

Formulas

corresponding

to the line

frequencies

and

intensities,

as well _as data relative to the relaxation

times,

can be found in the book

by Abragam

and

Bleaney _[15] (§5.3, §5.6, Fig. 10.5).

In the present

work,

this

analysis

is extended to the _case of NMR and Môssbauer

techniques.

A few formulas valid for H

#

0

(H

_m

r)

_are first derived. It is then assumed for

simplicity

that H

=

0,

1e. that the nuclear

spin

^is

only

submitted to a

magnetic hyperfine coupling:

7ii,f

=

Ajjlzsz ^(to

^which a

lattice-type quadrupole coupling

_q

(Ij @@j,

insensitive to the tunnel

effect,

_may be added if

desired). Indeed,

if S

# 1/2,

_as assumed

here, magnetic hyperfine

terms of the form

(Ai /2) (1+S-

₊

I-S+)

have _no matrix elements inside + S _>.

In what

iollows,

_we will therefore _use the notation A instead of

Ai

j.

The formulas

previously

obtained _are then used for

computing

the

irequencies

and line intensities

corresponding

to EPR and NMR

spectra

when 1

=

3/2,

and _to _a

simple

Môssbauer

~ transition

le

=

3/2

_-

Ig

=

1/2.

Such transitions exist, for

example,

in

5~Fe

or

~~~Tm [16].

2. General Formulas

Let _us start irom the Hamiltonian:

7i =

g~IBHSz

+

AI=Sz

+

r[S

><

-Si

+

ri

S _><

Si (3)

in which r is the

(usually

_non

adjustable)

tunnel

parameter.

^Let us assume that

Iz

₌ M. The

eigenvalue equation

Ieads to two energy Ievels:

EM

=

+/(g~IBHS

₊

_AMS)2

₊ r2.

(4)

The

tunnelhng splitting:

2/(g~IBHS

₊

AMS)2

₊ r2

=

/(2g~IBHS

₊

2AMS)2

₊ A2

(5)

is in

agreement

^with

equation (5.56)

oi

Abragam

and

Bleaney _[15]

for _an effective

spin

S

=

1/2,

and reduces _to A _e 2r when A

= H

= 0 _as

expected.

Let _us define:

~~~~~~~ (g~IBH/+ _AMS)

^~~~

in

which,

for

simplicity,

both numerator and denominator will be assumed to be

positive (see

Addendum at the end oi the

paper),

with:

cos(2ÙM

_"

~~~~~

~ ~~~

sin(2ÙM)

_" ^~

(7)

~/(g~IBHS

+

AMS)2

+ r2

/(g~IBHS

₊

AMS)2

₊ r2

There _are two

eigen

vectors

titi(M) _~2(M)

⁼

^COS(ÙM)lS,

^M ^> ⁺

^Sin(ÙM)1 ^S,

^M ^> ^j~~

=

sin(ÙM)(S,

M _> ₊

cos(ÙM)( S,

M _>

The first _one

corresponds

to E ₌ +

/7

and the second _one _to E

=

fi

in

equation (4).

When

(g~IBHS

+

AM) SP,

2ÙM _-

~,

_and:

2

l~i(M) _~2(M)

^-

^[~~

^>

^[M

^>, ^with

^E~,M

^" ^+r

-

[~~

>

[M

>, with

E~,M

_" -r.

When

(g~IBHS

₊

AM)

_»

r,

_2ÙM _-

0,

and:

l~i(M) _~2(M)

^-

^[S,

^M ^> ^with

^Es,M

^"

^+g~IBHS

⁺ ^AMS

-

S,

^M > with

E-s,M

_"

-gIIBHS

AMS.

Finally

<

i~i(M)lszli~iim)

_>

= <

1~21M)lSz14Î2(M)

>=

Scos(2ÙM) ₁₉₎

<iii(M)jSzj~2iM)>

=

-Ssinj2ÙM). (10j

From _now _on, it wiII assumed that H

= 0 and M _> 0. Then

EM

"

E-M

"

+~ _(ll)

tg(2ÙM)

"

~

,

cos(2ÙM)

_" ^~~~

,

sin(2ÙM)

₌ ^~

(12)

AMS

~ ~'

There _are two

degenerate

states with energy:

+~

~iimj

=

cosjÙmjis,

M _> +

sin(Ùmjj S,

M _>

iljj-Mj

~~~~

=

sinjÙmjjs,

-M _> ₊

cosjÙm)j S,

-M >

and two

degenerate

states with _energy:

-/r2

₊

(AMS)2)

IÎÎÎÎ~Î)

=

ÎÎÎÎÎÎÎÎÎ ~ÎÎ~Î~ÎÎÎÎ)[ _~),~ÎÎÎ~

>

~~~~

It what

iollows,

it wiII be assumed that the

(irreversible)

relaxation rates between aII the

states of the

problem

_are small

compared

to r and A. This _is

obviously

_necessary for the

observation of the

magnetic hyperfine

structure.

(5)

3.

Application

to EPR and NMR when 1=

3/2

It is assumed that the

hyperfine

Hamiltonian has the form:

~ ~~ ~

~~ ⁱⁱⁱ

⁺

)j

hf =

z z + q

z ~

and that S

(or

J for

R-E-)

is

larger

than

1/2 (for example,

J

= 6 for

Tb~+ ).

For I

=

(,

the four

lowest electronuclear _states _are four doublets with the

following energies

and _wave functions:

~

~ ~

4/i ((

₌ Cos

lÙ~/~) s,

+ sm

lô~/~) ^s,

~~~ ~~~~

~

~~ Î~

^~~~ ^~~~/~~

^~'

^~ ^~°~ ^~~~/~~

^~'

~

~ ^4/i ⁽⁾⁾

⁼ ^Cos

^lÙi/~) ^s, ⁾⁾

⁺ ^sin

^(ôi/~) ^s, ⁾⁾

~~~ ~ ~' ~~~~

4/[ (-j)

= sin

lÙi/~) ^s, -))

₊ _Cos

_(ôi/~) _s, -))

~

~ ^4/2 ⁽⁾⁾

⁼ ^sin

^lÙi/~) ^s, ⁾⁾

⁺ ^Cos

^(ôi/~) ^s, ⁾⁾

~~~~ ~ ~~~~

~ ^~~

~Î (~jj

COS

~~l/2) ^~,

Sl~

(Ù1/2) ~,

~

~~

^4/2

^(~)

⁼ ^sin

^lÙ~/~) ^s, ⁽⁾

⁺ ^Cos

^(ô~/~)

^s,

⁽⁾

~~~~ ~ ^~' ~~~~

4/i (- ₍₎

= Cos

lÙ~/~) s,

sin

lô~/~) ^s,

3.1. E-P-R- SPECTRUM. It _can be observed with _an RF field

parallel

to Dz:

~RF

=

gpBHisz Cos(~Jt).

The hne intensities _are

proportional

to

[sin(2ÙM

)]~

Line I

Transitions:

~2 ())

-

~i (() ~[ (-))

-

~[ (-)).

Intensity: II

_~c

[sm

(2Ù~

/~)]~,

^increases ^from ⁰ to 1 when r increases from 0 to _cc.

Fi.equency: huI

=

2~,

increases from 3AS to 2r +

~

~~~~.

4 4 r

Line II

Transitions:

~2 ())

_- _lYi

())1 ~[ (-))

_-

~[ (-)).

Intensity: III

_c~

[sm (2Ùi /~)]~,

^increases ^from ⁰ to 1 with r.

Frequency:

h_VII

=

2~,

increases from AS to 2r +

~~~~

4 4 r

These two EPR fines haire been observed _on

Tb~+

_in

Y.E.S.,

_as descnbed

by Abragam

and

Bleaney

_[15]

(Section 5.3,

Section

5.6, Fig. 10.5).

The

experimental

EPR

frequencies

_are:

huI

_" 14.932

GHz, huII

= 12.018

GHz, correspond-

ing ^to ^a

tunnelhng splitting

A

= 2r

= 0.387 cm~l. The relaxation rate is the _same for the two fines within the

experiniental

_precision, and at the lowest measurement temperature ^K.

the relaxation time _is about 25 _ms.

(6)

3.2. NMR SPECTRUM. It _can

only

be observed with _a R-F- field

perpendicular

to Dz:

~RF

_"

~"~"~~

(I+e~~~/~

+

I-e~~/~).

^The ^fine intensities _are

proportional

to the _square of 2

the matrix element

of1+ (3

^for ^-

~

,

4 for

- - +

).

2 2 2 2

Line 1:

Transitions:

~i (Î)

_-

i~i (Î)liÎÎ (~')

_~

~i (~Î).

Intensity:

~

_~ ~ ~~~~ ~~~~~ ~~~~~~~ ^~ ^~

~2 ₊

~~~~~~~

1 ~2 s2

~

~~~~~

4 4

Vfhatever

r/AS,

this

quantity

_never

departs

_very much from 3.

Frequency:

hui

_"

~-

A2S2 ₊

2q. (19b)

~2 s2

It _varies from AS +

2q

to +

2q

when r increases from 0 to _cc.

r Line 2:

Transitions:

q/ij 1j_q/ (lj~~~q/ jlj_q/ij _1j

2 ~à ¹ § 2 § 1 ~à'

Intensity:

Ajs2

~~~~~

12 _Cc 4 [COS

(2Ù1/2)1~

⁴ ~

~2 ₊

A(S~

It decreases from 4 to 0 when r increases from 0 to _cc.

Frequency:

hu2

" 2 A2S2.

(20b)

It varies from AS to _m 2r when r increases from 0 to _cc.

Line 3:

Transitions:

~2 (Îj

~

~2 (1)

~~~

~)

~

~~ (~Îl'

Intensity:

_same _as that of fine 1.

Its

frequency

derives from

hui by

the

change: +2q

_-

-2q.

~ÎOnCÎU310nT

. For r

=

0,

there _are three fines with:

hui

" AS +

2q, hu2

"

AS, hu3

_" AS

2q

and

intensities 3:4:3

(7)

. When r _- _cc,

hv2

C3

2r,

with

vamshing intensity,

and there remains

only

fines and

~2 s2 3,

with

intensity

3 and hu

= +

2q.

Line 2 and the

magnetic hyperfine

structure r

(h.f.s.)

of fines 1 and 3 _are

suppressed by

the tunnel elfect.

4.

Application

to a Môssbauer Transition

3/2

_-

1/2

The ~y transition _is assumed to occur between _an excited nuclear state

I~

= 3

/2

with _a

hyperfine

Hamiltonian

~hf

"

AI]Sz

and _a

ground

nuclear state

Ig

=

1/2

with

~hf

_"

-BIfSz ii-e-

trie

order of trie levels in trie

ground

state is

inverted,

_as in

5~Fe).

Trie energies and

eigenfunctions

of trie excited state with 1=

3/2

bave been

given

m trie

preceding

section. It will be assumed for

simplicity

that _q

= 0. Let _us define

~~~~~~~ BÎ~S' ^~°~~~~~

r2

ÎÎÎMS)2

'

~~~~~~~ ~

~~~~

The nuclear

ground

state

gives

_rise to trie two doublets:

~ ~

~ ^*2 ¹⁾⁾

⁼

^sin1~7i/~) ^s,

⁺ ^Cos

^l~Ji/~) ^s,

~~~~ ~ ~~~~

~

~Î (~ jj

COS

(§~1/2) ~,

_Sl~

~~'l/2) ^~, ~j

~

~ ^~i ^(j)

⁼ ^Cos

^1~7i/~) ^s,

⁺ ^sin

^(~Ji/~) ^s,

~~~~ ^~ ~

~[ (- j)

= sin

(~Ji/~) s, -j

~~~~

+ Cos

(~Ji/~) s, -)

For _a

magnetic dipole

nuclear transition between two states _a and

b,

and for _a

powder,

trie

intensity

is:

Iab

_ce

j<

_a

§ _>j~

+ < _a

§

_b

>

j~+ j< ai

^Ml~ jb

>j~ 124)

where Ml is trie

magnetic dipole operator.

It is well known that for transitions between

eigenstates

of

I(

and

If,

trie intensities of trie allowed transitions _are

proportional

to

3(3/2

-

1/2),2(1/2

_-

1/2)

and

ii-1/2

_-

1/2).

Here _we will

only

consider trie transitions between trie _vanous excited states and trie lowest

ground

state

Ah, 4l[ (first

half of trie

spectrum).

Trie second half of trie

spectrum

is

easily

deduced from it.

4.1. ASPECT _OF _THE

(HALF)

SPECTRUM

Line _a

~~~~~~~~~~

4Îl (Î~

_~

~1 (Î~' ~~ Î~

~

~~ Î~

Intensity:

~~~~~

^~

~

~2~~ _~2~~~~2

+

~2 _2~

~~~~~

(8)

It _varies from 3 to 0 when r increases from ₀ _to _cc.

Frequency:

h~ =

r2 ₊ ^~~2 s2 ₊

~(25b)

~

4

It varies from

)AS

₊

)BS

_to 2r +

) _~j~~

₊

) _~j~~

Line b

~~~~~~~~~~

fl~~

())

_-

~l (Î)Î

4~~

~~Îj

_~

Î~'

Intensity:

(àCOS (Ùl

_/2 §~l

/2)j

~

" Î + ~~

~ ~ ^~~

l. ^(~~~)

~~~~ ~~ _~

Î~~~~

It varies from 2 to 0.

~~~~~~~~~

hub

"

~

~~~~~

~ ~

~~~~~~ ~~~~ ~s ₊

)

~S tO 2r +

) _~~~

_~

Î ~é

Line ~y

~~~~~

4Îl (Î~

_~

Î~' ~~ Î~

~

~l ~~Î~

Intensity:

~~~~~~~~~ ^~ ~~~~~~

~

~2

~~~2~Î~~

~2 s2

l~~~~~

4

~

Always

_near _zero.

It varies from

)AS

₊

)BS

_to 2r ₊

) _~j~~

₊

) _~i~~.

Line _c

Transitions:

~l (~)

~

Îl' _ _Î~

_~

~l ~~Î~'

Intensity:

[Cos

iôi/~

+

~Ji/~ii

² ₌ +

~~

l) j[ j)~ )[ [j~~~1.

128ai

~ ~

(9)

Always

_near 1.

~~~~~~~~~

hUC "

~~~ ~

~~~~~

It _varies from

-)AS

₊

1BS

_to

-1~~s2

~ ⁱ B2s2

Line

fl

Transitions:

~

jlj

_q/

jlj,~ij lj_q/ij lj

ii ² à' 1~à 2 ~à

Intensity:

il

S1n (Ù1/2 ~71/2

Ii

~

" Î

l~

^~~

^ ^)~ ^~

~

j

(~~~)

~ _~ ~ ~ ~ _~ ~i ~

4 4

It _varies from 0 to 2.

Frequency:

hvp

₌ ^A2S2 + B2S2

=

hu~. (29b)

It varies from

)AS

₊

^1BS

to -1 A2s~

~ ⁱ B2s2

Line _o

Transitions:

~l')2Îl'')~~ Î~'

Intensity:

~~~~~

^~

~2~~ _~2~2 _2 _2~

~~~~~

It varies from 0 _to 3.

Frequency:

h~ = r2 ₊ ^~~2 s2 ₊ r2 ₊ f~2 s2

(30bj

°

~ ~'

It varies irom ^~ As ₊

)Es

_to ^~

^~~~~

₊

^~~~~

4.2. DIscussIoN

. For

arbitrary r/AS

trie half spectrum is

composed

of _six fines _a,

b,

_{~y, c,}

fl,

_a.

. Trie _sum of trie fine intensities _is

always equal

to

6,

_as

expected.

. When r

=

0,

trie half

spectrum

is

composed

of three fines _a,

b,

_c with

respective

^ire-

quencies hv~

=

~

AS +

BS, hvb

_" AS +

BS, hv~

= AS ₊

BS,

and

respective

2 2 2 2 2 2

mtensities

3, 2,

1

(this

is trie standard Zeeman

case).

(10)

. When r _m

AS,

BS

ii-e- (r/h)

m 50 MHz for

5~Fe,

m

10~

10~ MHz for

R-E-).

there _are three fines _a,

b,

_~y, with hu _m 2r and

vanishing intensities,

there _are three fines _c,

fl,

_a, with

respective frequencies hu~

=

hup

₌ ^~ ^~ +

~ ~ ~

~ ~ ~ ~

8

Î~

8

Î~

and

hua

=

-~

^~ ^~

+

~ ~

,

and with

respective

intensities

1, 2,

^3. ^In

practice,

8 r 8 r

since _uc

= up ^the

(half) _spectrum

is

composed

of two

equal

fines with

intensity 3,

at

frequencies

_u~ and _va

(note

that if A _m

B,

_va <

0).

Upon turning

no~v to the whole

spectrum,

it appears that when r increases from 0 to cc, its

aspect changes

from the standard six fine

spectrum

: a,

b,

_c,

c', b', a',

with fuit

magnetic hyperfine

structure and intensities 3 _: 2 1 1 2

3,

to a four fine

spectrum (c+fl),

_a,

a', (c'+fl')

with shrinked

magnetic hyperfine

structure and

equal

intensities 3.

In other

words,

the tunnel elfect _suppresses the

magnetic

h-f-s-. Note in

passing

^that in the presence of _a lattice

type quadrupole coupling,

and whatever

r,

fines _a,

a',

_a, ^a' will be shifted

by

+q, and fines

b, b',

_~y, _~y', _c,

c', fl, fl' by

_-q.

There exists _a

simple explanation

for the suppression of the

magnetic

h-f-s-- Let _us _con- sider for

example

the _case 1

=

3/2 (NMR

_or excited Môssbauer

state).

ivhen r _- _cc, the

eight eigenstates

_give rise to two electronuclear

quadruplets )~~

>

(M

> and

(~a

>

iii

_>

(M

₌

+), +)), separated by

2r. The electronic states

)~~

> and

)~a

_>

being diamagnetic,

these

quadruplets

have _no first order

magnetic hyperfine

structure. However there is _a

magnetic hyperfine

contribution to second

order, giving

rise to foui- doublets:

)~~

> +

3/2

> _:

E~

₌ r +

~

~/ ₍₃₁₎

~2 s2 )~~

> +

l/2

_>

:

E(

= r ₊

(32)

1

~2 s2 )~a

>

+1/2

_>

E[

= -r

(33)

)~a

> +

3/2

>

Ea

₌ -r ~ ~ ~

(34)

Since there exists _no nuclear matrix elements between

)~~

> and

)~a

_>, this enables to

explam

the NMR

spectra (disappearance

of _one

hne) and,

after extension to the Môssbauer

ground

state, ^the presence of four Môssbauer fines with

equal

intensities. This last

phenomenon

has _a little

analogy

with trie mechanism of the

pseudoquadrupole

interaction invoked for _ex-

plaining

the

quadrupole

spectra

of169Tm

_in

TmC13 6H20 ([16] §3.10

and

Fig. 3.9,

and ii. ii

).

Note that conclusions similar to those of the

present

_paper were derived _a

long

time ago for

an

analogous problem Ii?i (valence quadrupolar

structure of Môssbauer spectra in the _presence of Jahn-Teller

tunnelling

between two

wells).

At first

sight,

it does not seem very easy to find favorable _cases for the Môssbauer observation of the

shrinking

of the

magnetic hyperfine

structure due _to

tunnelling

^elfects.

^~5~Tb

is

unfortunately

net _a Môssbauer

isotope. 169Tm

is one, and

Tm~+

_has _J

= 6 _as

Tb~+;

how~

ever, smce the second order Stevens coefficients

o j of

Tb~+

and

Tm~+

are opposite, ^whenever

Tb~+

_has _a

)Jz

₌ +6 _>

ground

doublet

[4], Tm~+

^will

probably

have _a

singlet (or

at least

unfavorable) ground

state.

In orgamc or

biological compounds _[18],

there exist

examples

of ferric ions

5~Fe~+ (S

₌

5/2)

with

large

_spm Hamiltonians ~

=

-D'S]

and

ground

state doublets

)Sz

=

+5/2

>.

(11)

However,

for Kramers ions

[4], tunnelling splittings

_can

only

be induced

by application

of _a

transverse

magnetic

field

Hi,

with r _c~

HfS

c~

H[,

which is net very convenient. The _same

remark

applies

to

~6~Dy~+,

^J ₌

15/2,

whence r _c~ H

(5

for the

ground

doublets

)Jz

₌

+15/2

>

observed in _some

compounds

_[16] Tab.

17.8),

at least if trie level scheme is well

represented by ~o

₌

-D'J],

which is seldom trie _case in R-E-

compounds

_[4]. This is

elfectively

net trie

case for

l~°Yb~+,

J

=

7/2,

in

yttrium ethylsulfate,

ⁱⁿ which it bas _a

ground

state doublet

)Jz

₌

+3/2

>: for

predicting

trie variation of r with

Hi,

it will be _necessary to _use the exact

level scheme and _wave functions

(which

_are

fairly

well known

[19]).

In

Fe++

_ions

(S

=

2),

there is _an

important

orbital contribution to trie

magnetic hyperfine

structure

[20] [21].

^In

addition,

relaxation between orbital states is

usually

_very fast. There

are a few _cases of slow

relaxation,

but

apparently

without

tunnelling splitting

in trie

ground

state

[22].

5. Conclusion

The elfect of _a coherent tunnel elfect _on trie

magnetic hyperfine

structure of _a

single

_ion,

observed

by EPR,

NMR and Môssbauer

spectroscopies

bas been studied in _zero

longitudinal

field.

Explicit expressions

for trie fine

frequencies

and intensities bave been derived in the

case of EPR and NMR experiments

involving

_a nuclear

spin

¹ ₌

3/2,

and for _a Môssbauer transition

Ie

₌

3/2

_-

Ig

=

1/2.

Predictions relative to EPR had

already

been made and checked _a

long

time ago for

Tb~+

in

yttrium ethylsulfate, although

the _name "tunnel elfect"

was not used _at that time. The

present

paper

mainly

_surveys the

opportunities provided by

Môssbauer

spectroscopy

^for

observing

the

"shrinking"

of the

magnetic hyperfine

structure due to the coherent tunnel elfect.

A summary of these results has been

presented

as a post ^deadline poster at the International Conference _on the

Applications

of the Môssbauer

Elfect,

I.C.A.M.E.

1995,

Rimini

(Italy), September

1995.

Acknowledgments

The author _is indebted to Drs P. IMBERT and P. BONVILLE for useful discussions about

tunneling phenomena.

Addendum

In reference [4] ^{of the} present _paper, ^for orthorhombic

symmetry

and

integer S,

the

tunnehng splitting

2r is due to the term

~(S(

+

S~

e

2~(Sj S))

in

equation il

of [4]. ^{r is} given

by equation

là of _[4] for _even

S,

r is

always negative

and the

symmetric

state ~~~ ~

~~

~

~~

/à

is

lowest;

for odd

S,

r has the

sign

of

~~

_i-e- _trie

sign

of _~. However when _~ _< 0 trie _easy _axis in trie hard

plane

_is

Dz,

but when _~ >

0,

the _easy axis _is

Oy;

if Dz is chosen _as the easy axis

in the hard

plane,

_one must take _~ _<

0;

^then ^r is

always negative

and the

symmetric

state is lowest whatever the

parity

^{of S.}

Similarly,

in the

tetragonal

_case, trie

tunnehng splitting

_is due to trie term

u(S(

₊

S~

in

equation (58)

of [4]; this ^term is trie

"operator equivalent"

^{of the} ^classical

quantity 2u(S]

6SjS(

+

S().

For Dz _or

Oy

to be easy axes m the hard

plane,

it is _necessary to take _u < 0.

If _u _>

0,

^trie _easy _axes are

[l10][lÎ0].

r is

given by equation (59)

of [4]. ^When ^S =

4n,

r

(12)

is

negative

whatever trie

sign

of _u. When S

= 4n ₊ 2 and _u <

0,

r is aise

negative

and the

symmetric

state is lowest in bath _cases.

Similar discussions should also be carried out in the

trigonal

and

hexagonal

_cases.

In the present paper, we have taken r _> 0 for

simplifying

the

correspondence

between square root solutions such _as

equation (4)

and their limits when A

= H

= 0. As shown

above,

this may

correspond

to

taking

for Dz the hard axis in the hard

plane. Anyhow,

whatever the

sign

of

r,

the

physics

and the conclusions _are the _same.

References

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Scharf

G.,

Wrezinski W-F- and Van Hemmen

J-L-,

J.

Phys.

A

(London)

Math. Gen. 20

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[2]

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E-M- and Di Vicenzo

E-P-, Phys.

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10548 and references therein.

[3] Politi

P.,

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Hartmann-Boutron F. and Villain

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F.,

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il 5]

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A.

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and

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Chem.

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F.,

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