Interference effects in Mössbauer relaxation spectra

HAL Id: jpa-00208628

https://hal.archives-ouvertes.fr/jpa-00208628

Submitted on 1 Jan 1977

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Interference effects in Mössbauer relaxation spectra

F. Hartmann-Boutron, D. Spanjaard

To cite this version:

F. Hartmann-Boutron, D. Spanjaard. Interference effects in Mössbauer relaxation spectra. Journal

de Physique, 1977, 38 (6), pp.691-696. �10.1051/jphys:01977003806069100�. �jpa-00208628�

INTERFERENCE EFFECTS

IN

MÖSSBAUER RELAXATION SPECTRA

F. HARTMANN-BOUTRON

Laboratoire de

Spectrométrie Physique,

^B.P.

53,

38041 Grenoble

Cedex,

^France

and D. SPANJAARD

Laboratoire de

Physique

^des

Solides,

Université de Paris-Sud 91405

Orsay Cedex,

^France

(Reçu

^{le 16 décembre}

1976, accepté

^le

24 fevrier 1977)

Résumé. ²⁰¹⁴ Nous examinons l’effet sur les spectres ^de relaxation, de l’interférence ^entre la conver-

sion interne et l’effet

photo-électrique, qui

peut donner naissance à de petites anomalies dans les

profils

^{de raie.}

Abstract. ²⁰¹⁴ We investigate the effect in Mössbauer relaxation spectra, of the interference between conversion electrons and photo-electrons, which may

give

^{rise to} small line

profile

^anomalies.

Classification

Physics ^Abstracts

4.200 ^- 8.610 ^- 8.680

1. Introduction. ^- The recent

development

^of

very intense M6ssbauer sources

(several Curies)

combined with new detection

methods,

^{makes it}

possible

^{to obtain} spectra ^with

high precision (better

than 1

%) [1] (1).

^It then becomes _necessary to take account of small interference effects which are

likely

to

slightly

distort the spectra, ^{such as} the interference between conversion electrons associated with the nuclear

transition,

^and

photo-electrons.

^This

pheno-

menon has

already

been studied both

theoretically [2, 3, 4, 5]

^and

experimentally [6].

^The aim of the present

study

^{is to extend}

previous

^{formulas to the case where}

relaxation effects

(essentially paramagnetic spin-lattice relaxation)

^are present. ^We will therefore assume that the Mossbauer atoms are dilute

impurities

^{in a}

matrix.

As shown in ref.

[2],

the interference effects consider- red here are not observable in emission spectra ^for pure

multipolarity,

^{and as we shall see}

later, they

are not very easy ^to detect in the case of mixed mul-

tipolarity.

^On the other hand

scattering experiments

are

complicated (2). Consequently

^{we will be}

mainly

interested in transmission

experiments

i.e. in the

computation

^{of the} refractive index n. In the first

(1) Kalvius, ^G.

M. j

^{Shenoy, G. K., private} communications.

(2) An additional complication is the interference between resonant scattering ^from the nucleus and nonresonant scattering

from the atomic electrons.

part ^{of this} paper ^we will derive a

general expression

for _n,

allowing

^{for the}

mixing

^of

multipolarities.

^In

the second part ^we will discuss the

observability

of the interference terms and

give

^an

explicit

^formula

for the M6ssbauer

lineshape

^{in a}

simple

^case

(effec-

tive electronic

spin

^{S =}

1/2

and nuclear transition 0 +

2 + ). Finally

in the last part ^{we will}

give

^{the for-}

mulae relative to emission spectra.

2. General

expression

for the refractive index. ^- 2.1 NOTATION. ^- As discussed

by

^{Blume and Kist-}

ner

[7]

the refractive index is related to the _average value of the forward

scattering amplitude (ref. [7],

eq.

(9)).

^{In reference}

^[8]

^this

^amplitude

^{was calculated}

without

including

interference effects and for the

simple

^{case of a} nuclear transition with _pure multi-

pole

character. In such a case the conventions

adopted

for the tensor

operators

which describe the nuclear

multipole

^{moment are not} critical. On the contrary in the _presence of

multipolar mixing,

^{one has to be}

very careful. Several conventions exist in the litera- ture, those of Edmonds

[9],

Brink and Satchler

[10]

and Rose

[11],

^{and as an} illustration we

give

^{in the}

appendix

^the

expressions

for the inverse nuclear lifetime r ⁼

I/Tn

of the excited Mossbauer state

corresponding

^to all three conventions. Here we will

use the conventions of

Rose,

^{as was done}

by

^Hannon

and Trammell

[2, 3, 4]

^and

by

Blume and Kistner

[7].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003806069100

692

Let I and

Ig

be the nuclear

spins

of the excited and

ground

^{Mossbauer states and let us} define normalized tensor operators

Tr by

the relation :

These operators

satisfy

^the

orthogonality

^{relation :}

2. 2 EXPRESSION FOR THE SCATTERING AMPLITUDE IN THE ABSENCE OF INTERFERENCE EFFECTS. ^- Let us assume

that the nuclear transition I H

Ig

^{has a mixed}

multipolarity L,

^{L + I}

(in practice M 1 - E2)

^{and let us} compute the

scattering amplitude

in the absence of interference effects. If we allow for

polarization

of the incident and scattered _rays,

f

^{is a two}

by

^{two matrix.} The forward

scattering amplitude corresponding

^to circular incident

polarization p

and scattered

polarization p’ (p, p’ =

±

1)

^is

given by :

In this

expression k,

^and wi ^are the wave vector and

angular frequency

of the incident

photon (3), Ry

^is

the recoil free

fraction, 6 ^{e + iex}

^{is the}

mixing

ratio of the ^two

multipolarities

^{as defined}

by

Blume and Kistner

[7]

(when

time reversal invariance is

satisfied,

^{a can be}

only

^{0 or}

n). ’Dktp(t/JeO)

^{is a} rotation matrix and

(0, §)

^are

the

polar angles

of the direction of

k.

^with respect ^to the nuclear frame.

7L(t)

^{is a}

Heisenberg

operator ^{as in} ref.

[8b, 12],

whose time

dependence

^{is due to} the static

hyperfine coupling Jeo

^{and to} relaxation.

Finally a

^is

the Boltzmann

density

matrix of the

ground

^{nuclear state}

Ig.

^{Here we will assume for}

simplicity

^{that a = 1}

(JeolkB

^{T ,}

1). Eq. (3)

^{can be obtained}

by

^{an obvious}

generalization

of the calculation which leads from _eq.

(18)

to eq.

(34)

^of

[8a]

^and

(allowing

^for

changes

ⁱⁿ

notation)

it reduces to eq.

(34)

^of

[8a]

^for pure

multipole

^character

and

unpolarized light.

On the other

hand,

when the nuclear states I and

Ig

^are

^simply

^Zeeman

^{split by}

^{a static}

magnetic

^{field H with the}

geometry

^of

figure

^{1 of}

[7],

eq.

(3)

^{reduces to} eq.

(21)

^{of Blume} and Kistner. When

a ⁼

1,

eq.

(3)

^{can be cast into a more} compact ^form

by introducing perturbation

^factors

G mm’(t)

^{similar to those}

used in PAC

[13],

^{and their}

Laplace

transforms

6LL (P-)

2.3 INTRODUCTION OF THE INTERFERENCE EFFECTS. ^- This can be done with the

help

^of eqs

(4)

^and

(14)

of Hannon and Trammell

[4]. Eq. (4)

^of

[4],

^which

gives f

^{in the} absence of interference can be

put

^{in the more}

general form, analogous

^{to our} eq.

(3)

^{with a = 1 :}

(3) wi referred to the center of the Mossbauer spectrum ^{in the} absence of hyperfine coupling and of interference effects. We use the same

convention for co in eq. (18).

where

[(rf rf)1/2/r]

^exp

[i(q -L",L - 11-1)] represents

^the

mixing

ratio and where A ⁼ 0 for a

magnetic dipole transition,

^{and  = 1 for an electric}

quadrupole

transition.

It is then shown

by

Hannon and Trammell

([4]

^eq.

(14))

that the effect of interference can be introduced in _eq.

(6) by simply replacing :

where the small

quantities çi, çf,

^are

given by

eqs.

(16)

^and

(17)

^of

[4].

If we now go back to our own eq.

(5)

and consider the

specific

^{case of a} transition

EL+l/ML

^{the inter-}

ference effect will

modify

^{it to :}

It remains for us to

give

^the

expression

^of

GI"(p-)

^{in the} presence of

relaxation;

the derivation has been

performed

^elsewhere

[8, 14].

^{It is :}

where

JC’

^{is the}

hyperfine

Liouville hamiltonian

acting

^{inside I and}

Ig,

^S the relaxation

supermatrix [12, 14].

If) and I g >

^are electronuclear kets associated with the nuclear states I and

Ig.

If the incident radiation has

f F + F’

a Lorentzian

profile

^{centered at} WiO, with width

f,

^then

iWi - r

^{must be}

^{replaced by} ic0io - 2 ^[8].

2.4 RELATION BETWEEN

f

AND PHYSICAL OBSER- VATIONS. ^- The

relationship

^is

given by

^{Blume and}

Kistner

[7].

^Their eq.

(9)

^relates

f

and the refractive index n.

Their

_eqs.

(4-8)

^and

(5’-6’)

express the cha- racteristics of the transmitted beam as a function of the incident beam and n

(which

^{is a two}

by

^two

matrix).

3. Discussion. ^-

Spin

lattice relaxation studies

by

the M6ssbauer

technique

^have

mainly

^been per- formed in the Rare Earth _group. Almost all rare

earth

isotopes

^have

pure M1

^or

E2 transitions,

^with

the

exception of 153Eu, 167 Er and 171Yb

which have

E2/Ml

transitions. _{In eq.}

(8)

the interference terms in

ç

^are

present

^{both in the} pure

multipole

contributions

(first

^and

second’ lines)

^{and in the cross}

multipole

contributions

(third

and fourth

lines).

^{However we}

will see that the cross

multipole

contributions

disap-

pear in a number of cases and that when

they

^are

present ^the

computations

^{are much}

longer.

3.1 OBSERVABILITY OF CROSS MULTIPOLE CONTRI- BUTIONS. ^- It follows from _eq.

(8)

^{that the cross}

terms cannot be observed on a

powder

^{since :}

Also when the nuclear

spin

^is

coupled by

^an

isotropic coupling

^{with a true}

spin

^{S or an effective}

spin

^{S =}

1/2,

with

spherical relaxation,

^we have shown elsewhere

[14]

that :

(where &L(P)

^is

given by

eqs.

(28)

^or

(42)

^of

[14]).

Therefore the cross terms also

disappear.

Conversely,

^{in an}

applied magnetic

^field

parallel

^to

Oz the

Gt1¥(ff)

^{with L 0 L’ are not zero and the}

cross terms do not vanish. However the

interpretation

of

experimental

^{data will}

require

^some calculations :

as an

example

^for

171Yb, Ig

⁼

^1/2,

^{I =}

^3/2

^and

assuming

^an

isotropic

electronic

doublet,

^the

computa-

tion of the

Gt1¥(j’f)

^{in a}

^magnetic

field will involve the inversion of a 32 x 32 matrix. A similar calcula- tion

involving

^{a 36 x 36} matrix has

already

^been

performed

^{in PAC}

(in

^a

decoupled basis) [15].

3. 2 INTERFERENCE EFFECTS IN THE ABSENCE OF CROSS TERMS. ^- Let us now consider the much

simpler

case of a

powder.

^{In that} case,

taking

^{account of}

eq.

(10),

eq.

(8)

^{becomes :}

694

where in zero field the G can be obtained

by

^{the method}

of ref.

[14].

^In

particular

^{for a}

spin

doublet with

hyperfine

^structure

^AIS, Ag Ig

^{S :}

and :

where F and G are the

hyperfine

^{states associated}

with AIS and

Ag Ig

^{S and}

^Tis

is the electronic relaxa- tion time.

In the case of a pure transition

E2(0+ 2+)

^{as the}

84 keV transition of

I7°Yb,

^we thus obtain that

The transmission

signal

^{will be}

proportional

^to

Im

(n + + + n- -)

^{oc Im}

f

4. Emission

experiments.

^{- We will see that in}

emission

experiments

the interference terms

only

occur in the cross

multipole

^{terms and that}

they

^can

be observed

only

^with

polarized

radiation. Therefore

we must first establish a formula for emission of

polarized

radiation and then introduce the inter- ference effects.

4.1 M6SSBAUER EMISSION OF POLARIZED RADIA- TION. ^- The calculation can

easily

^{be done with}

the method of ref.

[16] by introducing

^a

polarization

variable

Cq

^{in the}

^starting hamiltonian,

eq.

(6) of [16],

which

becomes,

in the notations of that paper

(4) :

Then if we assume that the

temperature

^is

high enough

for reorientation effects to be

negligible ^{(Ro «} kB T, density

^{matrix of state I}

equal

^{to the unit}

^matrix),

we find that the total number of atoms

arriving

^{in the}

ground

^state

7g

^{per second after}

^emission

^{of a}

^photon

with _energy 1iro is

given by :

where,

ⁱⁿ the notation of _eq.

(6) :

As noticed

by

^Blume

[17]

the coefficient

Cp Cp* ,

in eq.

(17)

^{is an} element of the 2 x 2 matrix _p which describes the

polarization

^P of the emitted radiation :

and the emitted radiation with

polarization

^{P is}

given by

where M is defined

by

eq.

(18) (this

definition is

slightly

^different from that of Blume

[17],

eq.

(9),

but the results are the

same).

^{It is} easy ^{to see} that

(Mp,p),

^eq.

^(18),

is identical to

(lpp,)

^eq.

^(6).

^Therefore

in the absence of interference effects

(Mpp’)

^{will be}

proportional

^to

(fp’ p)

^eq.

^(5).

4.2 INTRODUCTION OF THE INTERFERENCE EFFECTS.

- This is discussed

by

Hannon and Trammell

[2].

Its amounts to

replacing :

Notice that this rule is different from that for scatter-

ing (eq. (7)).

^{Then in the presence} of interference effects we have that :

We see from this

equation

that the interference effects

are

only

present ^{in the cross terms. Then}

they

^cannot

be observed on a pure

multipole transition,

^{or on a}

powder

^{or on an}

isotropic

electronic doublet in zero

field. On the other hand

they

behave like the

phase

factor

eiiX.

Then the condition for

observing

^{them is}

the same as the condition for

observing

^{non time}

(4) ( 7T ) ^{in this} equation ^is equivalent ^to (A ⁺ 1) ^{of Hannon}

et al. (( x ) ^{= 1 for} magnetic dipole, ( x ) ^{= 0} for electric _qua-

drupole) ; q( = ± 1) ^{is the same as} our p.

reversal invariance which is discussed

by

^{Blume and}

Kistner

([7]

p.

422).

^{In the presence of a}

magnetic

field H with the geometry ^of

figure

¹

of [7],

^it

requires

that one observes the

quantity :

i.e. that one detects the radiation emitted with linear

polarization Pn

^{at ± 450 to the x axis}

(see [7]

p. 419 for

definitions).

It follows from

this,

that deviation from time rever-

sal invariance and interference effects cannot be

distinguished

^{in such an}

experiment.

^{On the} contrary in a transmission

experiment

^the

phase

factor and interference effects behave

differently

^{and could}

(in principle)

^be

separated [19].

FIG. 1. ^- Total transmission spectrum I(co), and interference contribution to it, AI(co) ( x 25), plotted ^{as a} function of hw/A

for different values of n/Ar1s.

5. Conclusion. ^- The central result of this _paper is eq.

(8),

^which

gives

the forward

scattering amplitude

for incident and scattered

polarization

^{p and}

p’,

in the presence of

mixing

^of

multipolarities

^{and of}

interference effects between conversion electrons and

photoelectrons.

This formula has been

applied

^{to the}

simple

^{case of an}

E2(0+ 2+)

transition in the _presence of

isotropic coupling

^{to an} electronic doublet and has

yielded

^an

explicit expression,

eq.

(15),

^{for the}

M6ssbauer transmission

signal.

^{In this}

^{case ç}

appears

only

ⁱⁿ transmission and could be obtained

by

compa- rison of emission and transmission

spectra.

These results are illustrated in the

figure

^which

represents ^the total transmission

signal /(w)

^for

monochromatic irradiation as obtained from _eq.

(15),

and the interference contribution to

it, AI(cv) (term

in ç

ⁱⁿ eq.

(15)), computed

with the numerical values

appropriate

^{to the}

r 7

doublet of

(170Yb)3 + (T /A

^{= 0.076}

^{3, ç = -}

^{1.7 X}

^10-2 [6a]).

^{It is seen}

that

AI(w)

^{is at most a few} percent of the total

signal.

Notice that in order to take account of the finite linewidth r’ of the source in actual

cases, p = f ia)i r

r + r’

in _eq. ^q

(15)

^{should be}

replaced

^p

by F + F’ -

^{Y P} ₂

^iWm.

This will not alter the

shape

^{of the curves.}

Acknowledgments.

^{- We wish to thank Pr. G. M.}

Kalvius for

calling

^{our attention to}

^this problem during

the M6ssbauer Conference in Corfu. We also thank Dr. R. A. B. Devine for

checking

^the

English

of the

manuscript.

Appendix.

^-

Expressions

for the inverse nuclear lifetime r ⁼

1/Tn corresponding

^{to the different}

conventions

existing

in the literature :

- Edmond’s convention

(ref. [16]

eq.

(25)) :

- Brink and Satchler’s convention

(ref. [18]

eq. (3.29)) :

- M. E. Rose’s conventions

(ref. ^[7]

^eq.

(A. 2))

for a transition

E2/Ml :

References

[1] VIEGERS, ^{M. P.} A., TROOTSTER, J. M., Nucl. Inst. Meth., ¹¹⁸ (1974) 257.

[2] HANNON, ^J. P., TRAMMELL, ^G. T., Phys. ^{Rev. Lett. 21} (1968) 726.

[3] TRAMMELL, G. T., HANNON, ^J. P., Phys. ^{Rev. 180} (1969) ^337.

[4] HANNON, J. P., TRAMMELL, ^G. T., Phys. ^{Rev. 186} (1969) ^306.

Notice that the definitions of x in ref. [2, 3, 4] ^{are all} different !

696

[5] KAGAN, ^Y. M., AFANASEV, A. M., VOITOVESKII, V. K., J.E.T.P.

Lett. 9 (1969) ^91.

[6a] WAGNER, F. E., DUNLAP, B. D., KALVIUS, ^G. M., SCHAL- LER, H., FELSCHER, R., SPIELER, H., Phys. ^Rev. Lett. ²⁸ (1972) ^530.

[6b] RUSSELL, P. B., LATSHAW, ^G. L., HANNA, ^S. S., KAINDL, G., Nucl. Phys. ^{A 210} (1973) ^133.

[7] BLUME, M., KISTNER, ^O. C., Phys. ^{Rev. 171} (1968) ^417.

Strictly speaking the R.H.S. of _eq. (9) of this reference and of our Eqs (3) (5) (8) ^{should contain a factor} 1/(1 ⁺ 03B1c) where _xc is the internal conversion coefficient.

[8a] HARTMANN-BOUTRON, F., ^J. Physique ³⁷ (1976) 537. In this reference _eq. (26) should read

f = - L3k 203C00127c

R03B3

diff > ^{= -} R03B3 Vdiff >.

[8b] HARTMANN-BOUTRON, F., ^J. Physique ³⁷ (1976) ^549.

[9] EDMONDS, ^A. R., Angular ^{Momentum in} Quantum ^Mechanics (Princeton University Press) ^1957.

[10] BRINK, D. M., SATCHLER, ^G. R., Angular ^Momentum (Oxford University Press, London) ^1962.

[11] ROSE, M. E., Elementary Theory of Angular ^Momentum (John Wiley, NY) ^1957.

[12] HARTMANN-BOUTRON, F., SPANJAARD, D., ^J. Physique ³⁶ (1975) ^307.

[13] DATTAGUPTA, S., BLUME, M., Phys. ^{Rev. B 10} (1974) ^4540.

[14] CHOPIN, C., SPANJAARD, D., HARTMANN-BOUTRON, F., J. Phy- sique Colloq. ³⁷ (1976) ^C6-73.

[15] CHOPIN, C., SPANJAARD, D., HARTMANN-BOUTRON, F., ^J. Phy- sique ³⁶ (1975) ^961.

[16] HARTMANN-BOUTRON, F., SPANJAARD, D., ^J. Physique ³³ (1972) ^285.

[17] BLUME, M., J. Physique Colloq. ³⁷ (1976) C6-21. We are

indebted to Dr Blume for sending ^a preprint prior ^to publication.

[18] ROSE, H. J., BRINK, ^D. M., Rev. Mod. Phys. ³⁹ (1967) ^306.

[19] HANNON, J. P., Nucl. Phys. ^{A 177} (1971) ^493.