A simple Derivation of the Tunneling Splitting for Large Quantum Spins

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A simple Derivation of the Tunneling Splitting for Large Quantum Spins

Françoise Hartmann-Boutron

To cite this version:

Françoise Hartmann-Boutron. A simple Derivation of the Tunneling Splitting for Large Quantum Spins. Journal de Physique I, EDP Sciences, 1995, 5 (10), pp.1281-1300. �10.1051/jp1:1995197�.

�jpa-00247136�

Classification Physics Abstracts

75.60Jp 73.40Gk 76.20+q

A Simple ^Derivation of trie Thnnefing Splitting for Large Quantum Spins

Flançoise

Hartmann-Boutron

Laboratoire de Spectrométrie Physique

(UJF-CNRS,

UA

08),

Université J. Fourier

/Grenoble

I,

B-P. 87, 38402 Saint-Martin d'Hères Cédex, France

(Received

8 June 1995, received and accepted in final form 6

July1995)

Abstract. With the help of

a perturbation treatment to lowest order, general expressions for the tunneling splitting have been derived for _a quantum spin S. This _{spm is} assumed to have _a main axial Harniltonian

(without

applied

longitudinal

magnetic

field),

and to be

submitted to perturbations with orthorhombic, tetragonal, trigonal or hexagonal symmetries, _or

to _a transverse field. The formulas obtained _are valid for general

(not

_very

small)

S. They _are compared with available results of the literature. Previous observations of tunneling splittings

in terbium compounds by EPR _are mentioned, with the relevant references.

Introduction

The

tunneling

^of _a

large single

quantum

spin

S is of theoretical interest _{as a} mortel for the

tunneling

of the

magnetization

of small clusters _or

nanoparticles

[1, 2]. Elaborate

approaches

have been

used,

such _as

mapping

^the

large

quantum spin into _a

partiale problem,

which is then solved with

help

of the conventional W.K.B. method for the

particle problem

_{[3], or}

resorting

to instanton

techniques (including

the contribution of the

Berry phase)

_[4].

In what

follows,

_we present _a naive

calculation,

based _on lowest order

perturbation theory,

for _{a spin} submitted to a main axial Hamiltonian

7io

"

-AS),

A _>

0,

and to _zero

longitudinal

field. The

tunneling splitting

2r is induced

by perturbations 7ii

of the

spin Hamiltonian,

with orthorhombic,

tetragonal, trigonal,

_or

hexagonal

symmetries, or

by

_a transverse field

(which

is the

only possibility

for half

integer

spin, because of trie Kramers

theorem).

To lowest

perturbation

order, r _can be

represented

_as the shortest chain

connecting

^the two

degenerate

states

(e.g.,

_(Sz

= +S

>).

It _can be

expressed exactly,

whatever S, _in terms of factonals. Use of the

Stirling's

formula transforms it into _a power of

"7ii/7io".

It has been checked that when _an exact solution does exist, the "factonal form" is identical _to the first

term of its _serres expansion as a function of

"7ii/7io".

On the other

hand,

in the _case of _a

transverse

field,

the "factorial form" and the

"power

form" coincide

respectively (exactly

_or

almost

exactly)

with results obtained

by

Korenblit and Shender [si ^and

by

Scharf _et ai. _[3]

by

use of different methods

(recurrence

relation and

W.K.B.).

© Les Editions de Physique 1995

It is mentioned that

tunneling splitting

has

already

been observed

(although

the _{name was} not

used)

in E.P.R. experiments _on trivalent terbium in

hexagonal yttrium ethylsulfate

[6,

ii

and in

tetragonal LiYF4, CaW04,

PbMo04 (8].

The detailed

organization

of the _paper is _as follows:

In _a first part, the

principle

of the chain method is

explained

_on the standard _case of _an orthorhombic

distortion,

with

explicit examples (S

=

4,

S

=

5);

^the "factorial form" of r for the

ground

doublet (Sz = +S _> is

compared

with exact

solutions,

^when available

(S

=

2,

^S ₌

3).

Stirling's

transformation

leading

to a

"power

form" is

introduced;

formulas for excited doublets (Sz = +m > are also

given.

The _case of

tunneling splitting

due to _a transverse

jietd

is then

considered;

the "factorial form" of r for the

ground

state doublet is

compared

with previous results of Korenblit and

Shender,

obtained

by

_a recursion

relationship

for determinants

[si,

the

"power

form" is

compared

with results obtained

by

Scharf et ut. with

help

of _a W.K.B.

treatment, ^{both for the}

ground

doublet and for the excited doublets [3]: the agreement ^looks very

satisfying.

Formulas _are also

given

for

tetragonat

and

trigonat

distortions. In the

he~agonat

case, there exists _a

simple example,

that of

Tb~+(J

=

6)

in Y.E.S. for which exact formulas

are available in trie literature [6,

ii. Generally speaking however,

search of

simple tunneling phenomena

in Rare Earth

compounds

raises

difliculties,

which _are listed and discussed.

The second part is devoted _to the variation

laws,

with

S,

of the parameters ^{of the}

spin

Hamiltonian of the

large spin

^{S. For}

this,

it is assumed that S is the _sum: S

=

£s~,

of N identical individual

spins

_s, with the maximum value S i

= Ns. The parameters of the

spin

Hamiltonian of S _are

expressed

in _ternis of those of the

spin

Harniltonian of _an individual

spin

s, with the

help

^of "transfer coefficients"

(analogous

to reduced rnatrix elements of the

Wigner

Eckart

theorem).

The "transfer coefficients" derived in this second part _are ^used in the hmit S - cc of the first part.

1.

Tunneling Splitting

1.I. ORTHORHOMBIC CASE

1.1.1. Justification of trie Chain

Approach by

Use of

Symmetric

and

Antisymmetric

Wave Functions. Let _us start frorn _a Harniltonian:

ii

= iio + iii _"

-As]

₊

vis(

₊

sfj Ill

with A _> 0, _(v( < A. The lowest state of

7io

is the

degenerate

doublet (Sz = +S _>. We _want to know whether its

degeneracy

^{is lifted}

by:

7ii ₌

vis(

₊

S~

=

2viSj S()

₌

2vO(iS)

₁₂₎

where

O(iS)

is _a Stevens operator

iAbragam

and

Bleaney iii,

Table

17,

_p.

867).

The selection rule àm

= +2 requires

integer

S. We _are not interested in S

= 1, ^{for which} the

degeneracy

is lifted _to first order _{in v,} and _we will _assume S > 2.

Even S. Let _us first

study

this _case, for

exarnple

S

= 4. We

only

need to consider the 5 x 5 submatrix of 7i

corresponding

to states (Sz = +4 >, (Sz = +2 > and

(Sz

₌ 0 _>. If

one uses the

symmetric

states (4s >,

(2s

> and (0 > on the _one hand and the

antisymmetric

states

(4a

>,

(2a

> on the other hand, 7ii has _no matrix elements between

symmetric

and

antisymmetric

states, _so it further separates into _a 3 x 3 matrix 1(0 >,

(2s

>,

(4s >)

and _a 2 x 2 matrix

1(2a

>,

(4a >).

The 2 _x 2 matrices (2s >,

(4s

> and

(2a

>,

(4a

> are

identical,

_so all

perturbation

terms constructed from them, ^shift the doublet

(4a

_>,

(4s

> as a whole. An

energy dilserence between

[4a

> and [4s ^> can

only

_appear ^when one builds

perturbation

terms going

through

_{[o >,} since these

only

exist for

[4s

>. The lowest order term going

through

_{[o >}

is:

~~ ^<

4s[7iiÎ2s

_><

2s[7iiÎo

><

o[7iiÎ2s

><

2s[7iiÎ4s

>

~~

(E4s E2s)(1~4s E0)(E4s E2s)

~

v~l< _4SISÎ

₊

S~

_{12S ><}

2SllSÎ

+ S~ )1°

>)~

1-A)3@2)(16)(12)

j~~

~~4~

^{~~ ~}

~/-

)3~~~jÎ

_{~ ~ ~}

~ _~

~)3 ^6~~[j2

^{~~ ~~ ~~~}

in which _we have made _use of the result:

j

s ₊

M)1

<

S,

M

kÎSiÎS'~

_~~

(5)

(S

₊ M

k)!

x

ÙW

whence:

<

-sjsisj

_{+ s >=}

_{j2s)1 jù)}

So state

[4s

> is

displaced by

2r with respect to

[4a

>

(tunnel splitting).

T

being negative, [4s

> is lower than

[4a

>. Note that T is also

equal

to:

~ ^<

-4j7iij

2 _><

-2j7iij

<

oj7iij

+ 2 ><

+2j7iij

+ 4 >

lE4 E2)lE4 Eo)lE4 E-2)

~~~

which is the shortest

perturbation

_term

connecting directly

^the

degenerate

states + 4 _>. Its

numerator has the form of _a chain: (4 >- (2 >- (o >- 2 _>- 4 _> between the two

states + 4 > and _{4 >,} which _can

easily

be visualized _{on a}

drawing,

whence the _name: "the

chain method".

Odd S. For S

= 1 the

degeneracy

is removed _to first order. Here _{we are} interested in

large spins,

in

practice

S > 3. Let _us

study

the _case S

= 5. We

only

need consider the 6 _x 6 submatrix of 7i

corresponding

to states + 5 >, ^{+ 3} >, +1 >. If _{one uses} the

symmetric

and antisymmetric states, ^it separates ^into two 3 _x 3 matrices

ils

_>,

_(3s

_>,

(5s

> and

(la

_>,

(3a

>,

(5a

>. These matrices _are identical except for trie

diagonal

matrix elements of

ils

>

and la >. Indeed:

<

ls(7iiÎls

>= <

la(7iiÎla >=<1(S((

-1>=<

-1(Sf(

_{+1 >=}

SIS +1) #

0

It then _seems

preferable

to rewrite the Hamiltonian _as:

7i=7i[+7i[

with

7i[

=

7io

+

ils

><

ls(7iiÎls

><

ls(

+

(la

_><

la(7ii (la

><

lai (8')

7i[

= 7ii

Ils

_><

ls(7iiÎls

><

ls( (la

_><

la(7iiÎla

><

lai (8")

An _energy diiference between

(5s

> and

(5a

> will

only

_appear when _one builds

perturba-

tion _terms

going through ils

> and

(la

_>, because of the diiference in the

energies

of the

denominators. The lowest order such

perturbation

term _is for (SS >:

,

<

5s(7iiÎ3s

><

3s(7ii Ils

><

ls(7iiÎ3s

><

3s(7iiÎ5s

>

~~~~

(E5s E3s)(E5s Els~

<

lS(~lÎlS >)(E5s E3s)

^~~~

or, since <

ls(7iiÎls

> <

(E5s Eisl, ôE(~

₌

ôE~

₊

ôE5s

with:

~~ ^<

5s(7iiÎ3s

><

3s(7ii Ils

><

ls(7iiÎls

><

ls(7iiÎ3s

_><

3s(7ii

_{(SS >}

~~

lE5s E3s)lE5s Eis)lE5s Eis)lE5s E3s)

^~~~~

~~55

~5

"

j_A)4 ^(~~)

<

-5(S£(

3 _><

-3(S£(

-1 ><

-1(S£(

+1 ><

+1(Si(

₊ 3 _><

+3(S£(

_{+ 5 >}

~ 16 _x 24 x 24 x16

(12)

~~~~

~ÎÎ]2

~

~~~~~

^{~~ ~~~~}

Similarly

_{one can} show that

ôE(~

₌

ôE~

₊

ôE5a

with:

ôE5a

" -r

State

(5s

> is

higher

_or lower than

(5a

>

depending

_on ^the

sign

of _v.

In this odd S case, r _can also be put ^{into the} chain form:

~ ^<

-5(7iiÎ

3 ><

-3(7iiÎ

-1><

-1(7iiÎl

><

1(7iiÎ3

><

3(7iiÎ5

_>

(E5

E3

)(E5

Ei

)(E5 E-1)(E5 E-3)

~~

1.1.2. Generalization

According

to the above

formulas,

r has the form:

<

-S(7ii1

_{S + 2 ><} -S ₊

2(7ii1

^S + 4 > < S

4(7ii ÎS

2 > < S

2(7ii ÎS

>

~

(ES ES-2) (ES ES-4). --(ES E-S+4 )(ES E-S+2)

ils)

It has

already

been noticed that the _numerator contains

12S)!

Since 7io

=

-AS),

the denom- inator _can also be

rearranged

_as follows:

(-A)~~~ _lS~ (S 2)~ils~ (S

_4)~i

_iS~ (S

6)~i

lS~ (-S

+

4)~ils~ 1-S

₊

_2)~i

=

(-A)S-1(2s 2)(2)(2s 4)

x

4(2s 6)

_x 6 x x 4 _x

(2s 4)

x

2(2s 2)

=

(-A)S-i

_x

225-2(s _i)

_{x i x}

_{(s 2)}

_{x 2 x}

_{(s 3)}

_{x 3 x x i x}

_{(s -1)}

~

~_ ~~s-i~2s-2j~~

_~~jj2

°~~~~~~°~~'

v~(25)!

(16)

~ _# ~chain

"

(_ ~)S-1

y

225-2[(

s 1)[j2

For _even S: ôE«

= 0,

ôEs

=

2r(r

< 0 because it contains

(-1)~~~

= -l.

For odd S:

ôE«

= -P,

ôEs

= +P

(where

the

sign

^of P is that of

v~,

_1.e., that of

v).

Remark: In Messiah's notations [9], ^{P coula also be considered} as the lowest order off-

diagonal

matrix element between two

degenerate

states:

<

-S(7ii ^~°

7ii

~°

.7ii

~°

7ii1

+ S >

i16)

a a a

in which

~°

=

L'~~

^~~

^~~

Matrix elements of this type _are

used,

for

example,

when

a

Es Em

the

degeneracy

of _a level is removed

only

to second order

ischiif, Quantum Mechanics,

2nd edition

[loi, Eq. (25.24) ).

^In our case the tunnel

splitting

coula also be considered _as

resulting

from the

diagonalization

^of _a matrix of the form:

(S

> S >

<

Si

-ôE P

(ii)

<

-Si

r -ôE

in

which,

to lowest

order,

r is _a chain

off-diagonal

matrix element.

l.1.3.

Comparison

witl1Exact Solutions

Case S

= 2. Let _{us measure} the

energies

with respect to that of the

unperturbed

doublet

(Sz

₌ +2 >. The

eigenvalue equation

_can be solved

exactly

with three solutions

(refered

to

the _energy E ₌ -4A of the

unperturbed doublet):

El

₌ 2A + 2A

fi

₊

12~ ⁽¹⁸⁾

(perturbed

_{energy of (0}

>)

E[

= 0

i19)

iperturbed

_energy of the

antisymmetric

state (2«

>) E( ~

= 2A 2A _{1 +}

12~ ⁽²⁰⁾

A

_1~~2 ~4 ~6

~~

~

A ~

~~A3 ~~~A5

^{~ ~~~~}

E(

is the

perturbed

_energy of the

symmetric

state

(2s

>:

The first term in _the expansion of

ôE(1-12v~

IA) is

identical to _2r btained by _the

chain

method.

_Note that the exact _solution is not derived from the

first

term of

ltenng

its

_numerical

by

theory

~ ~~~ _~ ~~ ~

~~~~~~~

igiving

the exact

E3s(-)

and Eis

(+))

~ ~~ _{~~ ~}

~~~

~~~~

igiving

the exact

E3a(-)

and

Eia(+))

The _exact value of the tunnel

splitting

of the lowest doublet is

given by:

E3s E3a _" 12v 4A ₊

(~

+ 4A

(~ (25)

For this _same doublet 3s, 3a _we have the _series expansions:

~~~ ~~

~~Î

^~

Î

~~ Î~ _~3 ÎÎ

~~

^{~ ~~~~}

45 315 405

~~~ ~~

~~A

_{4 A2 32 A3} ^~ _{64 A4} ^{~ ~~~~}

E3s _" E~ +

ôE3s,

E3a _" E~ ₊

ôE3s

with:

Ea~ ₌ -9A

-15~ () ~

+...

(28)

(shift

of the doublet _{as a}

whole)

~~~~

~~~~ Î ÎÎ

~ ~~~~

(splitting

of the

doublet)

The first term of

ôE3s

coincides with r~ha;n.

l.1.4.

Computation

of

lbnneling Splitting by

Perturbation

Expansion:

Case S

= 4. In this

case, there exists _an exact solution for E4a~

E4a ₌ -10A

~/36A2

+

(4Viv)2 ₍₃₀₎

~~~ ~~~

Î~

~

ÎÎ~ ÎÎÎ~

~ ~~~~

On the contrary,

E4s

is the solution of _a third-order

equation.

It _can be shown

by perturbation theory

that E4s _" E4a +

ôE4s

with:

ôE4s "

-35~

^~ _x

35~

+..

(32)

The first _term of

ôE4s

coincides with the value of 2r obtained

by

the chain method in the

beginning

of this

chapter.

The

origin

of the second _one is _more

complicated.

General

perturbation

expressions ^to order 4 and 6 _can be obtained

by

_use of Messiah's formulas [9]. ^If one assumes that <

0(V(0

>= 0 in Messiah's notation, ^{the 4~~ order}

perturbed

energy e4 is trie _sum of two terms

(Appendix):

e4 =< o

v~°v~°v~°v

o _{> <} o

v~jv

_o

>< o

v~°vlo

^>

a a a a a

The second _term contributes

equally

to E4a and

E4s(+196v~/27A~).

The first term, ^which contributes

only

to

E4s,

is the lowest order "chain term"

(2r

=

-35v~/A~)

obtained in

il.l).

On the other hand the 6~~ order energy e6 is the _sum of nine terms

iAppendix).

Trie 8~~ and

9~~ _terms contribute

equally

to E4a and E4s and

yield (-2744v~/243A~).

Trie 6~~ and fl~

ternis

yield

0. The first five terms contribute

only

to

E4s(-175v~ /6A~).

At first

sight,

with this standard

perturbation theory,

there does not _seem to exist any

simple

rule to derive the 6~~ order terms from the 5~~ order terms, etc.

Therefore,

for

general S,

the

tunneling splitting

obtained

by

the "chain method" will

only

be approximate, ^since

higher

order

perturbation

terms will be

missing; also,

even if

VIA

< 1, we will have _{no way} to check the convergence of the

perturbation

series when S _{- cc.} Other

approaches iinstanton

or WKB

methods)

_may be _more

approppriate

to this

limit,

but

they

_are much _more intricate.

It is nevertheless

reassuring that,

_m the _case of _a transverse field hi

inext chapter),

the chain method leads to formulas for all doublets

iground

and

excited)

which _are almost identical to those obtained

by

the WKB method [3]. ^The interest of the chain method rests in its

simplicity

and

adaptability ito phonon

assisted

tunneling problems

for

example il1]).

1.1.5. 7Yansformation

by

Use of

Stirling's

^Formula. We have found

that,

to lowest order in

),

_{the tunnel}

_splitting

_{of the}

_ground

_state

was

Es

E«

= 2r with:

~

i-1)S-Îx

_225-2 ^~

Îl (~~~ÎÎ!]2

~~~~

This _can be transformed with the

help

of the

Stirling's

formula: n! %

/ùn"e~"@

into:

r à

fils-ilÎS

_x

Ii ^ls

^~

^)~Î

^~

¹³⁴⁾

When S goes ^to

infinity:

ls _-1)

ji L)S

^~

^@

^~~~~

Whence:

4AS~/~

v S

4vS~/~

v s-1

~~~~° _~

l-1)~~~@ ~A) _l-1)~~~@ ~A)

^~~~~

in which AS and

)

_{must be}

_kept

_constant

_isee

_{Section 2}

on "transfer coefficients"

1.1.6.

ibnneling Splitting

of _an Excited Doublet + _{m >}

im

_>

0)

When _m

= 1 the

degeneracy

_is removed to first order in

vit

r~

v).

When _{m >} 1, it _can be

shown

by

the chain method that the tunnel

splitting

to lowest order _is

equal

to

2rim)

with

ito

lowest order in

)):

~i~)

"

(-1)mÎ122m-2

^~

im-1

^~

_ijm /1)!j2 ÎÎÉ ÎÎ! ^i~~)

When _m

=

S,

one recovers the _previous formula for

r(S).

Note that:

rjm)

m A _x

(1)~ j38)

with

(~

< l. When _m

decreases,

_{i e., one} climbs towards the top ^of trie

barrier, rim)

A

increases.

In

particular,

if _{m <} S

ivery

_near the top of the

barrier):

[(

⁺

jj(

~

s2m (39)

Then:

~~~~

~

(-l)m-ÎÎm

_1)!]2

ÎÎ

~~~~

When S _{- cc,} AS

= const,

VIA

₌ const,

but, ~(

will

finally

become

larger

than 1. Then 4

~

r

+~ const _x S~~~~ will become

comparable

to the distance between doublets (Sz = +m' _>

and

perturbation theory

will break.

On the other

nana,

transformation of the

general expression (37) by Stirling's

formula requires that both _m > 1 and

(S m)

> 1, _1-e-, the doublet + _m > is

far,

both from the lowest doublet and from the top of the barrer

(mid-height).

Then:

~~~~ i-1)~~Î2m~~~

_2àre2

1

1ÎÎ Î~

^{~ ~}

Î ^~

~~~~

Î~Î)m

Î~~~i~Î

~

ÎÎ~Î -ÎÎ2

^~

Î

^{~ ~}

Î

^{~~ ~} _~~~~

Because of the

assumption

_{on m,} this formula does not reduce to that obtained for _m

= S.

l,1.7. Remark. Because

ris)

_or

rim)

_{are powers:}

ris)

~

fi

m v

(j)~~~

m A

(j)~

_m A

()

^~~

₁₄₃₎

they

cannot

really

be put into _a

unique

activation form:

= exp

1-

^{~ with a}

^clearly

T To

kBT*

defined

prefactor 1/To.

1.2. TRANSVERSE FIELD CASE

1.2.1. Ground Doublet. Let _{us start} from trie Hamiltonian:

7i =

-AS)

₊

_h~(S+

₊ S-

(44)

in which h~

=

-~~~~~

< A. If S is _an

integer,

the chain _goes

through (S

₌ 0 _>. If S is 2

half-integer,

it goes

through

+ _>. The tunnel

splitting

is Es E«

= 2r with

(to

lowest 2

order _in h

IA)

h2S(~ s)j

h2S~s

~

(-A)2S~l[(25

_1)!]2

(-A)2S~l ₍₂₅

_1)! ^~~~~

For

integer

S: ôE« ₌ 0,

ôEs

₌ 2P _< 0. For

half-integer

S:

ôE«

₌ -P and

ôEs

₌ +r where P has trie sign of

h(~

=

hl~~~l,

_{1.e., trie}

sign of h~.

Upon application

of

Stirling's

formula:

~

(-l)~~~~

_x

/&e

(25 -1)AÎ (-l)2S~l/&@~

(25 -1)AÎ

^~~~~

When S _{goes to}

infinity:

Il

^{~~ l 1}

25

(2s)2s (1_

¹

₎₂₅ ⁽²⁵⁾²⁵

^{x e~l}

25

So the first expression for r becomes:

~~~°°

~

(~lÎ2~~~ÎÎ ÎÎÎÎÎ

~~~~

These

expressions

_are to be

compared

with those obtained

by

Korenblit et ai. [Si ^{with the}

help

of _a diiferent method. Korenblit et ai. _{use a} Hamiltoman:

7i=-DSj-H.S=-DSj-HszcosÙ-H~~+)~~~sinÙ

and with

help

of _{a recurrence}

method, they

find that to lowest order in

H/DS,

the tunnel

splitting

is

equal

to

(their Eqs. (6), (7)):

~ ~

~~~Î ^~ (2/~ _l)!

^~~~~

or, when S _{- co:}

~~

@~~

4DS ~~~~

With the substitution D _- A and

(-Hs1I1Ù)/2

_-

h~,

their A is

equal

to our

splitting

2r, except ^{for the}

sign:

A

= -2r.

However,

what

they really

obtain is A~: _see their

equation (A7).

Note that these authors

equally

compute ^{the matrix elements of} Sz and

S+:

within the symmetric

(I)

and antisymmetric

(II)

states when 2HS _cos 6 _< A

(see paragraph

below their

Eq. (16))

within the

perturbed

states _~fiI

= (S* > aI1d _~fiII

= S* > wheI12HScos6 _> A

(their Eq.

(18) ).

It _caI1 be showI1from their results that in this last _case, the tunnel eifect is

destroyed by

the

magI1etic

field componeI1t aloI1g ^the _easy ^{axis Oz. This will be reexamined} in _more detail

in refereI1ce

il ii.

1.2.2. Excited Doublet + _m >

(m

_>

0).

TheI1:

h2m

(s

+

m)1

~~~~

=

(-A)2m-1((2m 1)ij2

^~

(s m)1

^~°~

Since

~~

< l, the tunnel

splittiI1g

iI1creases when _one chmbs towards the top ^of the barrier.

A

If _m < S, 1-e-, _{very Ilear} the top:

(S

₊

m)! /(S m)!

£t

S~~,

_wheI1ce:

r(m)

à

~i~~m-ii~m _i~,i~ l~i~l

~~

(Si)

WheI1 S _{- co,} h~ ₌ const, ^AS ₌ coI1st ₌

C,

therefore

h~S/A

_~J

h~S~/C

will

fiI1ally

become

forger

thon 1. Then

r(m)

~J coI1st _x S~~~~ _will

diverge,

and

perturbation theory

will

break,

as

already

_seen in the orthorhombic _case.

On the other

hand,

transformation of the

general expression

for

r(m) by

_use of

Stirling's

formula requires that both _{m »} 1 and S _{m »}

(mid height

of the

barrier);

_one gets that:

~

~ ~y~~ 1

/~~~@fij

~~

s _{+ y~~} S+Î

~~~°~

(- l)2m-1

^~ 27re2

A(2m

₁₎₂ S _m ~~~~

which does Dot reduce to the result

already

established when _m

= S. After noticing thon when

m is

large

il/(2m 1)~i~~

-

e~/(4m~)~~

this leads to:

1 ~~(y~~ ¹

/~)

/~~~ s2 _y~~2 ^~~ s _{+ y~~} ^S+ Î

~~~~~

(-l)2m-1

^~ _7r ^~ 4Am2 S _m ~~~~

This is almost identical to the results of Scl~arf et ai.

([3], Eqs. (1.7, 1.8)

with ^~ ₌ h~ and 2

~/ =

A,

which _con be put ^{into the form:}

~

2m ~~ j

~ ~ =

~

=

~'Î~

_~~

s ₊ ~2

~ _~

2 ~ _~

(5~)

~ Tm 7r 8~/m2 ² S ₊ ^~ _m

Their solution _was obtained

by

_mapping the

problem

of _a

large

quantum _spin into _a

partiale problem,

which is then solved

by

_usiI1g the coI1ventional W-K-B- methods for the

partiale problem.

The differences between the two formulas _are the sigI1 ^factor in the first _one

(but

in the second _{one, >} 0

by definition)

and the presence of S + iI1stead of S inside the brackets

Tm 2

of the second oI1e. Scharf et ai. _are iI1terested in the hmit S _{- co, ~/ -} 0,

~/S

= constant, but

they

still retaiI1 _a

= ~/ S ₊ ^~

~

_{as a} parameter

(see

their discussion after

Eq. (2.19)

of

[3]).

The fact of

keepiI1g

S ₊ iI1stead of S has the

practical

iI1terest of

avoiding divergences

when 2

m = S.

When _{m »} 1, _m

# S,

the

1/2

_can be

dropped

in the brackets and the results of Scharf _et ai. and of the present work become identical.

When _m

= S in formula

(54)

derived from

equations (1.7-1.8)

of [3], ^{it becomes:}

AE~ =

~'~~@@

~~

($

_~~

(55)

~ '~

~~

When S _{- co:}

2S

+ -

(25)~~e~/~

2

~~

whence:

(AESÎ

_#

(~) 4~/SVÎ ()j

^~~

= 0.5248 _x

4~/SVÎ ()j

^~~

₍₅₆₎

~r 'f 'f

while Korenblit and SheI1der [Si, as well _as the present

work,

fiI1d:

~~~~~ ~~Î~ (~~ÎÎÉI ÎÎÎÎÎ

^{~~ ~'~~~~ ~}

_{(~~ÎÎ~l Î~ÎÎÎ}

^~~ ~~~~

The results _are

ideI1tical,

except for the small differeI1ce betweeI1 the two Ilumerical factors 0.52 and 0.56 whicl~

might

arise from diiferent

approximations

in the calculations.

1.3. TETRAGONAL CASE

It

corresponds

to the _case of the

ferrimagI1etic

^cluster "Mn12" with S

= 10

il,11,12].

1.3.1. Ground Doublet. Let us start from the HamiltoI1ian:

7i =

-AS)

₊

_~1(S(

₊

Si

=

-AS)

_{+ 2~1O(}

(S) (58)

where

OI (S)

is the operator

equivalent

of x~

6x~y~

+ y~ and _~1 < A.

The

degeneracy

of the

ground

state doublet (Sz = +S _{> can}

only

be lifted

by

the

OI

term with selection rule Am

= +4, if S is _even and S > 2. In the chair computation, we have to

distinguish

the _cases S

= 4n, ^{where the chain} goes

through

_{(0 >,} and S

= 4n _{+ 2}

(S

_>

2),

where the chain _goes

through

+ 2 _>:

However, _{as in} the orthorhombic situation, _we

finally

obtain _a tunnel

splitting:

E~ Ea

= 2r

with

(to

lowest order _in

)):

r -

~~i~~~~

~

~ ~~~ ^x

~ili~~

^x

^{~~~2Sjj,~~ (59)}

2

whence,

_upon

application

of

Stirling's

formula:

~ _y

8(s ~)@~-s-2

~s4 ^~~~

~

(-l)~/~~~Vi

A

)

)~Î

~~~~

~ _y

8(

s

~)@~-S-2

_~~s4 ^S/2

(~l)~~~~~~~@

~~A(S )~Î

~~~~

When S _{- co:}

ls4

^{S/2 s2 S i}

^{sS s2)S/2}

~

~S

~

(S

₂₎₂

(S 2) ji

^{2 S e-2 e-2}

S

whence:

8~s3/2

~~s2 ^S/2

~~~~7/2

~~s2 ^S/2-1

~S~°°

(_i)S/2-1 fi

(fij

(_i)S/2-1~2 fi

~2~ ~~~~

For S ₌ 4n + 2 or

S/2

₌ 2n + one has

(-1)~/~-~

= +l and r _cc ~1~"+~; the _sign of r is that of _~1. For S

= 4n _oI1e has

(-1)S/~~l

= -1 and r _cc -~1~" _< 0 whatever _~1. The

sigI1of

~lis

probably

related to the characteristics of the

tetragoI1al

distortion

(compression

_or

elongatioI1).

~s2 Note that when S

- co, the ratio

~

^{must be}

^kept

^constant

^(see

^{Section 2 on} "transfer coefficients" ).

1.3.2.

illnneling Splitting

of _an Excited Doublet (Sz = +m >,

(m

_> 0,

even).

Then

(to

lowest order _in ~1IA)

WheI1 _m

= S _{we recover} trie previous ^formula for

r(S).

WheI1 _m

decreases,1-e-,

_oI1e dimbs towards the top ^of the

barrier, r(m)

increases.

If _{m <} S

(very

_Ilear the top ^of the

barrier),

^{~~ ~}

^~~~

E£

S~~

_aI1d

(S m)1

r(n~)

ç~

16A

~s4

^m/2

(-1)m/2-1 [(q _1)lj2

^{~ 16A}

⁽⁶⁴⁾

~s2 ~s4 ^m/2

wh~~ s _{_ ~} ~s

= ~~~~t ₌ ~~~~t c

cm/2

_sm

' ' 16A ^'

16A~

Therefore

r(m)

~J const x Sm-~ _will become

comparable

to trie distance between doublets

and

perturbation theory

will break.

On trie other

bond,

for _{m »} 1 and S _m » 1

(mid height

of trie

barrier)

ii is

possible

ta transform trie

general

expression

by

_use of

Stirling's formula,

which

yields:

~~~°~ ~

(-

)m/2~~~

(m 2)

4111m ~Î21~~~

~

ÎÎ

^~~~~~ _~~~~

(which

does not reduce to the expression ^of

r(S)

when _m

=

S).

1.3.3. Orders of

Magnitude

of tl~e Parameters for tl~e Cluster "Mn12"

(S

=

10). Experi- meI1tally

A

~J 0.6 K [12]. ^A very

rough

estimate of ~lis to be fouI1d in refereI1ce [11]: let _us start from _a tentative value of the siI1gle ion coefficient

B( (B(

= 584 in

Abragam-BleaI1ey's notation,

_see

Eqs. (7.2a, 7.88)

of [7]):

10~~

cm~~

_<

(B((

< 5 _x

10~~

cm~~

(66)

and let _{us use} trie fact that in the

coupled

duster MI112, _~1~J

B( /2000

_[11]; ibis leads to:

o-à _x

10~~ cm~~

_<

(~1( < 2.5

x10~~ cm~~ (67)

wheI1ce _a tunnel

splittiI1g

in trie

grouI1d

state (Sz = +S _>:

0.4 _x

10~~

_{Hz <} ^~~ _{< 1.26 Hz}

(68)

in agreement with trie lower hmit of the relaxation lime in trie

high

temperature activated

regime: 1/(2irT)

~J 3 _x 10~~ Hz

[12,13].

However _{a more} detailed discussion _{is necessary} [11], because there exist various measuremeI1t

techniques,

aI1d because

parasitic

eifects

(Earth's

magnetic

field for

example)

_may

play

_a rote.

1.4. TRIGONAL CASE. In tl~at case, the tunnel

splittil~g

arises from _a term

OI

in tl~e spil~

HamiltoniaI1

(for

Rare Earths,

OI

could also

play

_a

rote). AccordiI1g

to Table 17 of

Abragam

aI1d

BleaI1ey

_{[7], p.}

869):

O(

=

(Sz(S(

+

St)

₊

(S(

₊

St _{)Sz (69)}

For

simplicity,

_we will work with the HamiltoniaI1:

7i=7io+7ii =-AS)+r(Sz(S(+Sf)+(S(+Sf)Sz) (70)

S must be _an

iI1teger

and 25 _must be _a

multiple

of 3. Therefore S

= 3n, ^S > 3. Because of the _preseI1ce of trie Sz operators, ^{trie chaiI1}

computation

is _more

complicated

thaI1 in the

precediI1g

_cases. After

adequate

factorizatioI1s aI1d

simplifications

of trie Ilumerator aI1d of trie deI1omiI1ator, trie final result is:

~~"~

~~~~~

AÎ~~Î~

1 ~

~25/3

~~)~

i),j2

^~~~~

3

Note that the denomiI1ator coI1taiI1s

l~ ⁾

^{iI1stead of ~~}

l)

!, as could be

expected

ai

3 3

first

sight.

After

applying Stirling's formula,

ibis becomes:

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

~~~Î21~~ ^e2 /Î~ ))1~~~~

~~~~

When S _{- co:}

1 1

s

3)25/3

^~

^s2S/3~-2

Whence:

~~~~~'~~°° ~~~~~

~~Î~ ÎÎ

^~~~~

~~~~

Accorjjng

^{to the}

^properties

^{of traI1sfer} coelIicieI1ts

(see

Section

2),

when S _{- co,} AS

= coI1st.

~~~

A

~°~~~'

l.5. HEXAGONAL CASE. The creation of _a tunnel

splitting by

_a term of the form:

O(

₌

(J(

₊

J~ (74)

2

looks

possible

for

f

ions like Rare Earths. In

general however,

the

simple

model based _on

7io

_"

-AJ)

is _Ilot

applicable

to R-E-

(see

^discussion in Section

1.6). Nevertheless,

there exists

a favorable _case, that of Tb~+

(J

₌

6)

in

yttrium ethylsulfate,

where the chaiI1 calculation is not Ilecessary, since _an exact

diagonalization

in the basis

(6s

_{>, (0 >,}

(6a

> is

possible

and

is indeed

presented by Abragam Bleaney

_[7] (§ 5.3,

Eqs. (5.18-5.22);

5.6,

Eq. (5.56) ).

The tunnel

splitting

A

= 0.387 cm-~ has been measured

by

E-P-R- _{in zero} de

field,

with _an RF field

parallel

to the

crystal

axis

(Abragam Bleaney

_[7], 5.6, Tables

5.9-5.10).

The

longitudinal

relaxation lime has also been measured

(Abragam BleaI1ey

[7], 10.5,

Fig. 10.5). Although

the

Ilame "tuI1nel eifect" is _non

used,

this

experiment

is

quite

similar to thon

performed

on ferritin

by

Awschalom _et ai.

[14-16].

Another favourable _case

might

be thon of Terbium diluted in

Y(OH)3,

if such _a

compouI1d

_caI1 be

prepared (data

of the literature _are relative to coI1ceI1trated

Tb(OH)3 (17]).

Note thon for terbium in

Y.E.S.,

the eifect of its

hyperfiI1e coupliI1g

7in

=

Ajjszlz

with its

owI1 nuclear spin 1

=

3/2,

is the _{same as} thon of _a

magnetic

field

g~~IBH

=

Ajjm.

In _zero externat

field,

there _are therefore two tunnel

frequencies given by Abragarn

and

Bleaney

_[7],

Eq. (5.56)),

for _an effective spiI1 Se~r ₌

1/2:

&w

=

((A(fm)~

⁺

^{A~ ~~~ (75)}

WheI1ce,

in terms of the _{true spiI1:}

hw

=

((2AjjmS)~

+

A~

_)~~~

(75')

AccordiI1g

to the captioI1 ^of

Figure

10.5, these

frequeI1cies

_are:

u2 # 14.932 GHz for _m

= +

vi " 12.018 GHz for _m

= +

These

frequeI1cies

_are

high eI1ough

to avoid _aI1y deleterious eifects of other Iludear

spins (which

are

probably

rather far from

Tb3+).

On the other

hand,

relaxation rate measuremeI1ts _were carried oui oI1ly dowI1 to Kr- 2u2 _+~ 3ui, which is too

high

to observe _aI1y

possible

saturation of

1/T,

ai very low temperatures

Il il.

1.6. SPECIFIC PROBLEMS RAISED BY THE SEARCH _FOR TUNNEL EFFECT IN RARE EARTH

COMPOUNDS.

GeI1erally speaking,

ail the terms of the

crystalline

electric

potential

_con-

tribute to the

eigenvalues

and the

eigenfunctions

of the R-E- _ion. Therefore ils levels caI1I1ot be described

by

_a

simple

HamiltoI1ian 7io ₌

-AJ)

aI1d the _wave fuI1ctions _are mixtures of the

form _~fi

=

~j

_o

jM

(JM

>

M

For non-Kramers ions, the

ground

state is not

Ilecessarily

_a doublet.

WheI1 the

grouI1d

store _{is a}

doublet,

ii is Ilot

necessarily (Jz

₌ +Jmax > _or

(Jz

= +Jmjn >:

for

Yb~+(J

=

7/2)

in

yttrium ethylsulfate,

it is +

3/2

>.

When _a non-Kramers doublet

corresponds

to an irreducible representation ^of dimension two of 7io ₊ 7ii, _or to two

degeI1erate

oI1e-dimensional irreducible representatioI1s

(case

of the

r3,4

doublet of

Ho3+(J

=

8)

in

LiYF4),

its

degeI1eracy

_caI1

only

be lifted

by

an externat

magnetic field,

_as for Kramers doublets.

When the tunnel

splitting

is due _{to a}

perpendicular

field

hi,

_in order _to have _an eifect iI1volviI1g ^{virtual excited} states, oI1e must bave gi = 0. If trie eI1ergy levels

happeI1

to be well

described

by

7io

=

-AJ),

_theI1A

= 2r

~J

h[~ (for example hf~

^for J ₌ là

/2 (Dy3+, Er3+),

which is Ilot very convenieI1t for

experiments).

In order _to bave moderate

A(< cm~~)

and small admixture of trie excited stores into trie

grouI1d

state

doublet,

it is Ilecessary that the excited stores be

high eI1ough (>

100

cm~~).

Hyperfine

structure eifects _are for from

beiI1g Ilegligible,

with

coupliI1gs

of trie order 10~ -103

MHz,

aI1d there _are 100%

~~~Pr, ~~~Tb,

~~~Ho, ~~~Tm. WheI1

available,

isotopes without

Iluclear

spiI1would

be

preferable (Dy, Er, Yb).

Up to _now

[7,18],

trie

only

reliable observations of tunnel

splittiI1gs

in R-E- ions _{seem to} be:

Tb3+(J

=

6)

ⁱⁿ

hexagonal

YES: A

= 0.387 cm-~

([6,

_7] and Section

1.5) Tb3+(J

=

6)

in

tetragonal

LiYF4~ A

= 27.98 GHz _[8]

in

tetragonal

CaW041 A

= 8.1 GHz [8]

in

tetragonal

PbMo04~ A

= 15.8 GHz _[8]

In other _cases, distributions of

crystal

distortions _are preseI1t

(LaC13~Tb~+ [19],

YV04 aI1d similar matrices [20]

). FiI1ally, accordiI1g

to Môssbauer

Spectroscopy,

there _{exist some}

dyspro-

sium

compouI1ds

with _a

grouI1d

store + là

/2

> [21], ^but information _on trie excited states is

lackiI1g.

2. Transfer Coefficients

2.1. AIM _{OF THE} CALCULATION. Wl~eI~ _many spil~ s; are

coupled

to

give

_a resultant spiI1

S,

trie traI1sfer coelIicieI1ts relate trie terms of trie spiI1 HamiltoI1iaI1

7i(S)

of trie

coupled

system, to trie terms of trie spiI1 HamiltoI1iaI1s

~j7i,(s;)

of trie iI1dividual

spiI1s

_s,. TraI1sfer coefficients _are

proportioI1al

ta the reduced matrix1 elemeI1ts of the

Wigner

Eckart theorem

([22], Eq. (5A.1)).

Here,

_we wiII be iI1terested in the _case of N identical

spiI1s

_s aI1d of _a maximum resuItaI1t spin S

=

~j

s, with S

= Ns. This wiII

give

us trie way in which trie parameters

A,

h~, _{u, ~1,}

1

r, iI1troduced in Section 1, _vary, wheI1 S

(1.e., N)

_goes ta iI1fiI1ity.

The operators

Or

in what follows _are defined in agreement with

Abragam

and

Bleaney ([7], Appendix B,

Table

16). They

_are

proportional

to

(Tz

+

Tpm)

_or to

(Tz Tp~),

where the

Tz

_are operator

equivalents

of the

spherical

harmonics.

According

to trie

WigI1er

Eckart

theorem,

for giveI1 _n, ^{the reduced} matrix elemeI1ts of the components

Tz

of _an irreducible

tensor operator Tn _are the _same whatever _m. Ii is ofteI1 _{easier ta} compute them for _m

= 0.

2.2. FIRST-ORDER TERMS

(COUPLING

_TO A

FIELD).

This _case is very

simple:

~h.s;=h.~js,=h.S ₍₇₆₎

; ₁

The traI1sfer coefficient

relatiI1g

Ls, to S is

equal

to 1 and h must be

kept

constant wheI1 S _{- co.}

2.3. SECOND-ORDER TERMS

2.3.1. Axial Case. We _are interested in the transfer coefficient for:

~~i,

= B

~j3sÎ, 8(s

₊

_i)1 (77)

; _,

in which:

3s), s(s +1)

=

O( (s,),

is _a component ^of an irreducible tensor operator ^of order 2

(O(

= operator

equivalent

^{of 3z~}

r~). According

to trie

Wigner

Eckart theorem:

B

£ O((s,)

=

bO((S)

_or:

;

B

~j[3s(, s(s +1)]

=

b[35) ^S(S

+ 1)]

(78)