Remarks on the study of spin glass dynamics by Mössbauer and muon techniques

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Remarks on the study of spin glass dynamics by

Mössbauer and muon techniques

F. Hartmann-Boutron

To cite this version:

F. Hartmann-Boutron.

Remarks on the study of spin glass dynamics by Mössbauer and

muon techniques.

Journal de Physique Lettres, Edp sciences, 1982, 43 (24), pp.853-859.

�10.1051/jphyslet:019820043024085300�. �jpa-00232135�

Remarks

on

the

_study

of

_spin

_glass

_dynamics

by

Mössbauer

and

muon

_techniques

F. Hartmann-Boutron

Laboratoire de

_{Spectrométrie Physique}

_{(*), USMG,} BP 53 X, 38041 Grenoble Cedex, France

(Reçu le

_{30 juillet}

1982, accepté le 29 octobre _{1982 )}

Résumé. 2014 Du fait de leurs échelles de _{temps différentes,} la

_{spectroscopie}

Mössbauer et la

_précession

des muons

_positifs

fournissent des

_{renseignements complémentaires}

sur les verres de

_spin.

Dans les

verres de

_spin

_classiques,

des

_expériences

«

longitudinales

» en _champ nul sur les muons ont été

interprétées

en termes de ralentissement de fluctuations de

_{grande amplitude;}

nous montrons _que ceci conduit à des _temps

_{caractéristiques}

au

_voisinage

de

_TG

_qui

ne _{paraissent pas}

_compatibles

avec les

résultats Mössbauer.

Abstract 2014

Because of their different time _scales, Mössbauer _spectroscopy and

_03BC+

SR _give

comple-mentary information on

_spin

_glasses. We show that for standard _{spin glasses} in zero _field, the

inter-pretation of

_longitudinal

muon

_experiments

in terms of a _slowing down of

_{large amplitude}

fluctua-tions, leads to characteristic times around

_TG

which do not seem

_compatible

with Mössbauer

results _(**).

Classification

Physics

Abstracts

75.40 z 76.90 - 76.80

There is

_presently

some

_uncertainty

about the exact nature of the

_{spin glass}

transition. Is it similar to a true

_phase

transition

_(assumption

_~))

or is it due to a

_{progressive slowing}

down of

_{large amplitude}

fluctuations of

magnetic

clusters

_(assumption

(B)) ?

Information about the

dynamics

of

_spin

_glasses

can be obtained

_by

various

_methods,

_{among which} Mossbauer and muon

techniques play a

central role. However the

_{interpretation}

of the results which

_{they provide}

requires

some care : the aim of the

_present

_paper is to draw attention on a few

_{important points.}

1.

_Dynamics

and static effects in MÖSSbauer

_spectra.

2013 11 DYNAMIC EFFECTS 2013 We

_only

consider here the Mossbauer effect

of 57Fe

_(I

=

3/2 -+ Ig

=

1/2,

Ey

= 14.4

keV).

It is well known

(*) Laboratoire associe au C.N.R.S. _{(LA 08).}

(**) The

subsequent

text is an _expanded version of a communication (8 G ₉₎

presented

to the International

Conference on _{Magnetism, Kyoto (1982).}

L-854

that when the Fe nucleus is submitted to a static

_hyperfine

field

_H,

the Mossbauer _spectrum of a

powder

is

_composed

of six lines : a, a’

(external), b,

b’

(median),

c, c’

(internal)

with relative

intensities

_{3, 2,}

1. What

_happens

if H

_begins

_jumping

between two

_opposite

and

_equally

_probable

values

_HZ

=

+ H ?

At each

jump,

lines aa’

exchange

their

frequencies

and

similarly

for

bb’,

cc’.

If the

_jumping

rate Q =

lite

is small

compared

_to the Zeeman

splitting

the _spectrum is

unchanged,

being

the

_{superposition}

of two identical

_spectra.

If Q

_increases,

lines cc’ broaden and

_collapse,

followed

by

bb’ and

lastly by

aa’. At the same time a line

begins

to _grow at the centre _{of the}

spec-trum, and for a fast

jumping

rate the _spectrum reduces to this

_unique

line

_{(extreme narrowing}

limit).

Computed

spectra based on this model _may be found in reference

_{[1] :}

the intermediate zone

corresponds

to values

_{of ! c comparable}

to the

_hyperfine

circular

_frequencies,

i.e. :

!e ’" 5

x

10-9 S - 10 - 8

s. The same

_qualitative

evolution is observed for a

_fluctuating

H of

constant

_length

which reorients

_randomly

in _any direction in _space

_[2].

On the

_contrary

let us now assume that

_{H jumps}

between several values

_HZ

=

HI, H2, H3, ...

which are

_unequal,

with

_{unequal probabilities}

_{Pl I P2, P3.} For small

_jumping

rates the

_spectrum

is the sum of several different Zeeman _spectra associated with

_HI,

_H2,

_H3...

Then as Q increases the _spectrum broadens and

_collapses

and

_finally,

in the _very fast relaxation

limit,

it

consists,

not of a

single line,

but of an unbroadened Zeeman _spectrum

_{corresponding}

to the _average field

H ~

=

PI

HI

+ _P2

_H2

+ _P3

_H3

... This is what Mossbauerists call an « effective field situa-tion ».

Therefore an unbroadened Zeeman

_spectrum

_may

correspond,

either to a field

jumping slowly

between

_{equal values,}

or to a field

_{jumping rapidly}

between

_unequal

values

(or

_{alternatively}

to

H =

H >

₊

6H(t),

where the fluctuations

6H(t)

are fast

_compared

to the Larmor

_period

associated with the _average

_value

_{H )).}

A

_typical

_example

of an effective field situation is encountered in standard

_magnets,

where the

_hyperfine

field is

_proportional

to the

_magnetic

moment :

_H,,

=

A

S ~

(in

this case

_6H(t)

A

_~S(t)

_corresponds

to

_spin

wave

_modulations,

whose

_frequencies

are much

_higher

than

_{hyperfine frequencies).}

The relation

_H,,

_{= A}

_{S )}

is

obeyed

even _very near

_TN

[3] : comparison

with NMR results in

_{MnF2 [4]}

indeed _suggests that for

_{57 Fe,}

critical effects must be smaller than the Mossbauer

_linewidth,

due to the smallness of

hyperfine coupling

constants.

Finally

it should be noted that for a

_fluctuating

field whose

_jumping

rate varies from slow to

fast with

_temperature

_(as

in

_{superparamagnets}

_[5]),

the _aspect of the ME

_spectra

_changes

dras-tically

on a

relatively

small

_temperature

interval.

1.2 STATIC EFFECTS. - Because the Mossbauer transition takes

place

between nuclear states

with different

_{gyromagnetic factors,}

lines

_cc’,

_bb’,

aa’ also

_correspond

to different

_gyromagnetic

ratios

_(I

_Yee’

_{I I}

_Ybb’

_I

_I

_Yaa’

_I)

_(1).

_Therefore,

if the nuclei of the

_sample

are submitted to an

inhomogeneous

static

_hyperfine

_field,

all ME lines will be

_{inhomogeneously}

_broadened,

but with

increasing

widths from cc’

_(internal

_lines)

_to

aa’

_{(external lines).}

Such

_spectra

are

observed,

for

example,

in

_amorphous

_magnets.

’

1. 3 SITUATION IN STANDARD SPIN GLASSES. - It

should first be noticed that in non _{very dilute}

spin glasses,

Mossbauer spectra above the «

_{spin glasses}

transition » are often

_{perturbed by}

quadrupole

effects

(associated

with

_pairs

of

_neighbouring

_atoms)

which should not be

_interpreted

as « _{short range} order ». When these effects do not exist

_(RhFe)

_[6]

or are eliminated

by

dilution

(AuFe) [7],

above

_7~

the ME

_spectrum

_{(Fig. 1)}

consists of a

_single

line which broadens as T

decreases. It then

_splits

into two wide

_wings,

two small lines _appear near the centre and

pro-gressively

a full Zeeman structure

_develops.

The fact that lines _{cc’ appear} first and also that in

Fig. 1. - 57 Fe Mossbauer

spectra of Au 1.7 % Fe (estimated

T G

= 14.8 _± 0.5 K). The small shoulder

on the left in the 17.4 K _spectrum is _part of a

quadrupole

doublet associated with residual _pair effects. After

Violet and _Borg, Ref.

_[7].

the full _spectrum the linewidths increase from cc’ to aa’ is

_{strongly suggestive}

of an « effective field

situation »

_(case

_{OA )}

in _{the presence} of

_{inhomogeneous}

broadening

(on

the _contrary, for

_slowing

down fluctuations

(case

l~6~)

aa’ _{would appear first} and cc’

last,

the

_aspect

of the _{spectrum would} be different and would

_{change rapidly).}

When T continues

_decreasing,

the _average Zeeman

splitting

grows

(as

in a

ferromagnet)

while the line

broadening

diminishes. This _may be

interpreted

in the

_following

_{way :} because of

_disorder,

the different

_magnetic

atoms i are submitted to different

exchange

fields

_H~(~),

whence a distribution of the

lengths I S(i) > I

of their _average

_spins

S(i) >

and a distribution of effective

hyperfine

fields

(I

_H,,(i)

_oc

I ~ S(i) ~ ~ 1).

When T

decreases,

I

H~(~’) ) ~ )

(averaged over i)

increases and so do

_{~( S(i) ~ ~ ~}

and

_{H,,(/) ~ ).}

However when T -+

_0,

because of

_{saturation I S¡ > I --+}

S

_{whatever I}

_{Hex(i) > 1 whence}

the line

narrow-ing.

Mossbauer

_experiments

therefore seem to _support

_assumption

61’

Note however that in more

_complicated

_systems

_(amorphous

or

_ternary

_compounds)

it _may not be

_possible

to choose between static and

_{dynamic interpretations.}

2.

_Dynamic

and static effects in muon

_techniques.

- Let us recall that muons have a lifetime zu - 2.2 x 10 - 6 s and a

_spins

=

1/2,

with _a

gyromagnetic

ratio

y~/2

7c = 13

kHz/G.

In solids there are two _types of muon

J1. +

_{experiments :}

- transverse

precession

experiments

in an

_applied

field

_{Ho [8],}

which are similar to NMR free

precession experiments

and measure a

_quantity

_Gx(t)

oc

_{s,,’,(O) s,,-,(t),}

-

longitudinal experiments,

with or without an

_applied

field

_Ho

_[9],

which measure

G,(/)

oc

_sl,,’(0)

4(t).

In

_spin

_glasses

both

_techniques

are

used,

with an observation time of the order 1 ~.s to several _~s and a resolution of the order 50 ns to 100 ns.

L-856 JOURNAL DE _{PHYSIQUE -} LE’ITRE~

At low

_T,

muons _occupy fixed interstitial

_positions

a

and,

in dilute

spin glasses,

they

are sub-mitted to a field

H(a)

_{approximately}

_given

_by

_{[10] : H(a)}

=

H

~~ +

"dip(a),

in which

H)) ~

Ho

and

_Hdip(LX)

is the field with

_dipolar

_symmetry created at the muon a

by

the

_magnetic

moments

of its

_{neighbours ia}

in a Lorentz

sphere

(typically,

at concentrations c ~ 1

_{% , ~}

_{I Hdip(a)}

_~100

_Oe).

Both

_length

and orientation of

_Hdip(a)

_depend

on a. If the

spin

glass

is

_composed

of

_magnetic

clusters with

_{large amplitude}

fluctuations

_(assumption

then

the

_length

of

_H~;p(a)

should

correspond

to the full

_magnetic

moments and be

_temperature

_independent,

while its orientation should

_depend

on time with some characteristic time _LC

_{(dipole-dipole anisotropy}

is

_neglected

for

_simplicity).

On the _contrary, within

_assumption

₀

_(phase

_transition),

in which the time

_independent

_average value

₍

_{H~p((x) )}

is related to the

₍

_{S(ia) )’s}

of the

neigh-bours,

while

_~H~p((x,

_t)

is associated with their

_rapid

fluctuations

_5S(~,

t)

(spin

wave

excitations,

etc.)

which do not influence the muon

₍

effective field situation

_{»). Because}

of disorder

_H~p(x)

varies with (x, but the average over a of its

_length,

I

Hdip( a) ) I ),

should be

_proportional

to

the _average

_{magnetization}

₍

_{S(i) ) E )}

and increase with

decreasing

temperature. Both with

assumptions

A

and

l~6~,

the distribution of

_lengths

associated with

_positional

and directional disorders

shou d

be a Gaussian in the concentrated case and a Lorentzian in the dilute case

[ 11 ],

characterized

_by

their widths A.

Therefore two

_{possible explanations}

should be examined for

_interpreting

muon

_{depolarization}

in

_spin

_{glasses :}

- either

a time

_independent

_average

_{Hd,p(x) ~}

with a distribution of width

which increases when T decreases

_(assumption

_~~),

-

or a

fluctuating

_Hdip(a)

with a variable

_jumping

rate

_v(T)

=

_1/r~(T)

that decreases with

temperature, but with a

distribution

of constant

_(maximum)

width J f"too; S

_(assumption

®).

With their _present accuracy, transverse muon

_experiments

are not able to

_distinguish

between static effects

_(X)

and

_{dynamic damping}

_{~B .}

_{Although they}

are also sensitive to

_{inhomogeneous}

broadening, longitudinal

muon

_experiments

in zero field have been asserted to _support the

dynamic assumption

(B)

but this deserves careful examination.

First on the theoretical side

_{[9a,12,13] :}

in the static

_limit,

both Gaussian

(Fig.

_2a)

and Lorent-zian

_{(Fig. 2b)}

distributions lead to

_decreasing

_{G.,(t)’s (with}

different

_departures),

which _go

_through

a minimum when td w 1.8 before

_tending

to

_{1 /3}

as t -~ _{oo ;} on the

_contrary

in the fast relaxation

limit,

only

with the Gaussian distribution does one recover the extreme

_narrowing

effect

_(damped

Gz(t)

with a

_damping

constant f"too; J 2 Iv f"too;

¿j2

_rj.

Various

_recipes

had to be devised in order to

Fig. 2a. -

Computed longitudinal depolarization

function

_Gz(t,

_{1, v),} as a function of (At), for a _fluctuating

MUON RELAXATION FUNCTION

G~G (t)l.N SPIN

GLASS

Fig. 2b. -

Computed longitudinal depolarization

functions

_{Gz ~(t,}

a, v) as a function _{of (at)} in the Lorentzian

dilute case, as obtained

by

Uemura et _al., Refs.

_[9b,

_9c]. This is the result of a calculation in which the averages

over the orientations of the

_spins

and the

_positions

of the

_spins

are carried out _separately. The static case is

represented by

the dot dashed line. In the notation of the present paper, a should be

replaced by

d.

obtain the

_narrowing

effect even in the Lorentzian

(dilute)

case

[9~ 13].

When this is achieved it

appears that the sets of curves

_{GZ(t) corresponding}

to

_assumptions

@

_(J

7, v =

0)

(2)

and

(B)

(fixed

A,

v~)

look _very similar

_except

at

_long

t. With

_0,

_Gz(t)

_goes

_through

a minimum when

td ~

2

and then tends to

_{1 /3}

when t -+ oo. With

at

low

_v/J,

Gz(t)

exhibits first a minimum and then a

hump

before

tending

to zero; at

high

v/J,

Gz(t)

goes

monotonically

to zero with

decreasing damping.

Now when one looks at the

_experimental

curves for

_Gz(t)

in standard

_{spin glasses}

below

_{T~ [9d]}

_]

and

_figure

_3,

their

_present

statistics do not seem

_good

_enough

to

_clearly

choose between

_(X)

and

(no

unquestionable

minima

and,

all the _more,

_humps,

can be

_seen).

However most authors

Fig. 3. -

L-858 JOURNAL DE _{PHYSIQUE -} LEITRFS

have

_interpreted

their

_experiments

on the basis of

_assumption

(B). In

this _way

_they

find a

_rapid

slowing

down

_(v~)

when T decreases below

_T~,

with a characteristic fluctuation time _Tc around

7~

of the

_{order r‘ ~}

10-8

s in

_CuMn,

_AgMn

and AuFe

_([14]

_Fig.

_2).

But if this

_happens

to be true, Tc

being comparable

to the usual Mossbauer

_{hyperfine periods,}

ME _spectra of AuFe at

_{T G}

should

_{rapidly develop}

a full

_hyperfine

structure

_(similar

to what is observed in

_{superparamagnets).}

This does not occur at all and this

fact,

together

with the

_{satisfactory interpretation}

of ME

_spectra

in terms

_{of inhomogeneous broadening}

is not in favour of

_assumption

With

_assumption

_@

the

_{increasing damping}

below

_T~

is

_simply

accounted for

by

the thermal variation of

obiously

this does not eliminate the

_possibility

of a critical

slowing

down of certain excitation

modes near

_T~

as in standard _magnets, but information on the excitation _spectrum at low

fre-quency is

lacking.

To

_summarize,

in the

_present

status of both

theor and

experiment,

muon

_damping

in standard

spin glasses

in zero field can be attributed either

(A)

to an

_increasing,

non

_fluctuating,

_average

Hd,p{x) Bthe

deviations of

_Hd;p(x)

with

_respect

to this _average

_being

too fast to influence the muon or

B to

a

fluctuating

_"dip(Cl)

of fixed

(maximum) length

whose fluctuations slow down. The

_fluctuating

time at

_{T G}

derived with

_assumption

does

not seen to be

_compatible

with the Mossbauer _spectra of

_5’Fe,

which are better described in terms of an

_increasing

effective

field,

together

with a

_{large inhomogeneous broadening,}

in _agreement with

_assumption

O .

In fact with muons the

_only

safe _way for

discriminating

between

_dynamic

and static effects is to

_{perform longitudinal experiments}

in the _presence of an external

_decoupling

field

_H

_Hd;pXI

[ 15] :

this field _suppresses static

_damping,

however it also reduces

_{dynamic damping}

_{([15], Eq.}

₍₁₁₎₎

and

_{probably perturbs}

the

_{spin glasses.}

-A final conclusion is that the situation in

_{spin glasses}

is far from

_being

_understood,

as shown

_by

the

_generalized

use of

_Log-Log

_plots

in the literature and

_by

the

_comparison

of informations

derived from different

_techniques

_([17],

_{Fig. 1).}

Acknowledgments.

- I am indebted to Dr. P. Monod for

_calling

_my attention to this

_problem.

Note added in

proof.

- Uemura et al. have

_{recently performed longitudinal}

_{,u +}

_experiments

on CuMn in the presence of an

_applied

field

_larger

than the

_dipolar

field. A

_strong

reduction in the

damping

of

_GZ(t)

is observed

_immediately

below

_7~;

this has been attributed to the _emergence of a

quasi-static

_{component of S,}

giving

rise to

_decoupling

effects similar to those

_expected

in the

presence of an _average

_A(T)

under

_assumption

(X).

However an additional

fluctuating

_part

bS(t)

with a

_{fairly long}

characteristic time _{r~ ~}

10 -

-10 - 9

s is still introduced into the

_theory

(see

also

_[14,

_16]).

More details about these

_experiments

and about the various models elaborated in reference

_[9]

can be found in Y. J. Uemura’s

thesis,

as well as in the communications 3 G 14

(Uemura et al.)

and 7 E 2

_(Yamazaki

et

_al.)

to ICM 82

_Kyoto.

I am

greatly

indebted to Dr. Uemura for

_{communicating}

his thesis.

References

[1] WICKMAN, H. H., Mössbauer

_{Effect Methodology,}

vol. _{2, p. 39,} I. J. Gruverman Ed. _(Plenum

_Press)

1966. See

_Fig.

6, p. 55.

[2]

DATTAGUPTA, S., Hyp. Int. 11 (1981) 77. See

_Fig.

_{5, p. 105.}

[3]

WERTHEIM, G. K., Mössbauer

_{Effect Methodology,}

vol. _{4, p. 159,} I. J. Gruverman Ed. _{(Plenum Press)} 1968. See

_Fig.

2, p. 164 ;

Figs.

9-10, p. 174-175.

[4] HELLER, P., in Critical Phenomena, Proc. of a _Conference,

_{Washington (1965)}

_{(Ed. by} M. S. _Green, J. V. _Sangers, NBS

_Misc)

1973.

[5]

VAN DER _KRAAN, A. M., J. _Physique Coll. 32 _{(1971) C1-1034,} ICM Grenoble. In real _{superparamagnets} there is a distribution of 03A9 due to different

_grain

sizes.

[6] WINDOW, B. et al., J.

_Phys.

C 3 ₍₁₉₇₀₎ 2156. See _Fig. 3.

[7] VIOLET, C. E. et al.,

Phys.

Rev. 149

₍₁₉₆₆₎

540. See

_Fig.

2.

[8] MURNICK, D. E. et al.,

Phys.

Rev. Lett. 36 ₍₁₉₇₆₎ 100.

[9] a) HAYANO, R. S. et _{al., Phys.} Rev. B 20 _{(1979) 850;}

b) UEMURA, Y. J. et _al.,

_Phys.

Rev. Lett. 45 (1980) 583 ; c) UEMURA, Y. J., Solid State Commun. 36 _{(1980) 369 ;}

d) UEMURA, Y. J., Hyp. Int. 8 ₍₁₉₈₁₎ 739.

[10] MEIER, P. F., Hyp. Int. 8 ₍₁₉₈₁₎ 591.

[11] WALSTEDT, R. E. et al.,

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