MÖSSBAUER SPECTRA IN THE PRESENCE OF ELECTRONIC RELAXATION IN A QUADRUPLET Γ8 OF RARE EARTHS

(1)

HAL Id: jpa-00216714

https://hal.archives-ouvertes.fr/jpa-00216714

Submitted on 1 Jan 1976

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.

MÖSSBAUER SPECTRA IN THE PRESENCE OF

ELECTRONIC RELAXATION IN A QUADRUPLET

Γ8 OF RARE EARTHS

C. Meyer, F. Hartmann-Boutron, D. Spanjaard

To cite this version:

(2)

JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 12, Tome 37, Décembre 1976, page C6-79

MOSSBAUER SPECTRA IN THE PRESENCE OF ELECTRONIC RELAXATION

IN A QUADRUPLET r

8

OF RARE EARTHS

C. MEYER, F. HARTMANN-BOUTRON

Laboratoire de Spectrometrie Physique, BP 53, 38041 Grenoble Cedex, France and

D. SPANJAARD

Laboratoire de Physique des Solides, Universite Paris-Sud 91405 Orsay Cedex, France

Résumé. — Nous étudions l'influence de la relaxation à l'intérieur du quadruplet r% sur le spectre Môssbauer des ions de terres rares. Ce problème a été traité récemment par Sivardière et Blume qui ont utilisé un modèle stochastique. Nous considérons ici le cas plus réaliste où la relaxation est due aux électrons de conduction d'un métal. Le calcul est fait pour une transition nucléaire 0+ 2+ et pour le quadruplet r* le plus général, qui est décrit par quatre constantes P, Q, m et n.

A titre d'illustration nous donnons les spectres de relaxation correspondant, d'une part à un vrai spin f, d'autre part au quadruplet r8 de (Yb170)3+.

Abstract. — We study the influence of relaxation inside the A quadruplet on the Mossbauer spectra of rare earth ions. This problem was treated recently by Sivardiere and Blume with the help of a stochastic model. Here we consider the more realistic case of relaxation by the conduction electrons in a metal. The calculation is done for a nuclear transition 0+ 2+ and for the most general r8 quadruplet, which is described by four constants P, Q, m, n. As an illustration relaxation spectra

are obtained, first for a true spin j and second for the A quadruplet of (Yb170)3+.

1. Introduction. — In a recent paper Sivardiere

and Blume [1] studied the effects of electronic relaxa-tion inside a quadruplet T8 on the Mossbauer spectra

of rare earth ions. It is well known that inside r8 the

matrix elements of J are not simply proportional to those of an effective spin S = f, but are of the form : Jx = aSx + bSx and so on.

The theory of Sivardiere and Blume amounts to applying to S the stochastic relaxation model of Scherer-Blume [2, 3]. We show elsewhere [4] that this model does not necessarily correspond to a situation of physical interest.

On the other hand, even within the approximation of spherical relaxation, the most general relaxation of a true spin S = § is described by three independent parameters (correspondingtok = \,k — 2,k = 3) [4]. And for a quadruplet T8 in cubic symmetry we need

at least six independent parameters

(]c = i -> r

4

, k = 2 ^ r

3

+ r

5

,

k = 3 -• r2 + r4 + r5) .

In order to reduce this number, we shall here investi-gate the case of a Mossbauer impurity in a metal, where the electronic relaxation is due to the coupling

of J with the conduction electrons, which we will assume to have the simple form (*) :

3&t = -27>st(gj-l)J.s. (1)

The electronic relaxation supermatrix R' corres-ponding to r8 is a 16 x 16 matrix. We will see that

within the assumption of spherical symmetry of the bath (2) it only depends on one relaxation parameter.

We will also have to consider the hyperfine super-matrix JCo. We shall take as an example the case of a transition 0+ 2+ (Sm152, Dy1 6 0, Er1 6 6, Yb1 7 0, Yb1 7 2)

for which the hyperfine matrix of the excited electro-nuclear state is 20 x 20. Then the total matrix which determines the Mossbauer lineshape is an 80 x 80 matrix.

2. Properties of the quadruplet T8. — The properties

of the quadruplet T8 are reviewed in the book by C1) In principle this is not the most general expression for this coupling but it is widely used, and has the advantage of introduc-ing only one parameter.

(2) This assumption is valid only at temperatures which are high compared to the hyperfine structure. Our theory is therefore restricted to this case.

(3)

C6-80 C. MEYER, F. HARTMANN-BOUTRON AND D. SPANJAARD

Abragam and Bleaney 151, p. 721-732. As already supermatrix R' and the magnetic part Je, = Aj I. J

mentioned, the matrix elements of J inside

r,

are of the hyperfine interaction will involve P and Q . related to those of a fictitious spin

2

=

9

by two On the other hand, the hyperfine quadrupolar inter- constants a and b, or P and Q ([5] eq. (1 8 .16)). The action as given by eq. (1 8 .50) and (1 8.5 1) of [5] matrices of J, and J+ in terms of P and Q are given depends on two other parameters, m and n, and may by Sivardiere and Blume [I]. Both the relaxation be put into the form (see [5] eq. (17.14)-(17.24)) :

Note in passing that in cubic symmetry there is no quadrupolar contribution of the lattice, that the antishielding factor associated with

XQ

is not very big (-- 20

%)

and that it is fairly well known [5 bis]. This might be of interest for determining nuclear quadrupole moments (for example that of SmlS2). Also one must not forget that the four parameters P, Q, m and n are not independent. Indeed, even when

T,

appears several times in the decomposition of J in a cubic field, the expressions of the kets associated with

r,

only depend on one parameter x, which is the ratio of the sixth and fourth order components of the cubic field (Lea Leask and Wolf [6]). Therefore, once x is fixed, P, Q, m and n are determined and can be obtained from relations such as eq. (18.38) of [5]. In practice if the hyperfine parameters are known, x can be deduced from the low temperature spectra where relaxation effects are negligible.

3. Electronic relaxation supermatrix R'. - In order to obtain the electronic relaxation supermatrix R' we write X, as

and

where the matrices of V,, U+, U-, W3-, W3+ a re easily obtained from eq. (16) of Sivardibre and Blume [I].

The relaxation supermatrix is then deduced from eq. (51) of [7] in which we make the assumption of spherical symmetry of the conduction electron thermal bath (i. e. Jq(o) = const. independent of q and w, where Jq(o) are the spectral densities associated with the components of the conduction electron spin s :

Finally it turns out that

where

is the relaxation rate that would characterize a spin

S = 4, and R" is a 16 x 16 matrix whose elements will be given now. For this, let us denote by v, p the eigenvalues of S, and label the kets ( vp

>

in the following order :

(4)

4. Matrix of the magnetic hyperfine interaction. -

The magnetic hyperfine interaction

involves terms of the form

w w N

Iz

sz,

I +

s-,

I -

s+

w

which commute with I,

+

S, = F,, and also terms of the form I ,

5:,

I-

5:

which connect states for which F,

-

Fi =

+".

It is theref2re convenient to classify the~lectronuclear states (I, S) according to the value of S,

+

I, = p

+

m. Here we consider the case of a nuclear transition (O', 2 9 , i. e. there is a hyperfine structure only in the excited nuclear state 1 = 2.

Let us label the kets

I

pm

>

in the following manner

By virtue of the symmetry of the matrix with respect to its two diagonals we only need 18 matrix elements. If.

A

we set X, = -! X' these matrix elements are : 4

On the other hand it is well known that for rare earth It follows that ions

A,/h = 303 MHz

AJ = 2/3g,Pn

<

r - 3 >4f

<

J l l

NII J >

.

₍₇₎

(8) in agreement with hyperiine structure measurements (see [51,eq. 526). As an for the level _{on the doublet}

_r7

_using_{EPR or the M&sbauer effect} of (Yb170)3 + :

1101 (A,, = 3 A,).

gn pn I = 0.67 P n ([91

P.

611)

<

r - 3 >4f = 12.50 _{U. A. ([5] p. 300)} 5. Matrix of the quadrupolar hyperhe interaction. -

(5)

C6-82 C. MEYER, F. HARTMANN-BOUTRON AND D. SPANJAARD

are not quite convenient for our purpose. Let us v .-,

define : N =

((;/J,J+

+ ~ + ~ , / f ) ) / Z f i

(11)

B j =

-

e2

Q,

<

r-3 > 4 f

1(2 I

-

1)

<

J

11 11

J

>

(9) and set

Je,,,.,

=

9

4 B j 32". We use the same labelling

as for the magnetic hyperfine matrix. Here too the M = ((;13J:-J(J+ 1) (10) matrix is symmetric with respect to its two diagonals,

and we only need 22 elements :

Je;,

= 2 M Xi, = 2 J j ~ XI;,

= J ~ M + J Z N

X ' ; , , ~ = J Z M - J Z N

X ' =

-

M X 5 =

f

i

~

X';, = @ M f

f

i

X2,16

~

= @ M - J ~ N Xg3 =

-

2 M X ; , = O

x:,,,

=

JZ

M

+

JZ

N

x;,,,

=

J Z

M

-

.\lZ

N xi;, =

-

2~

x;,

=

-

J Z N

3 q , = M X = 0 X1& = - 2 M X;,,, =

-

2 f i N X1;, = - M X =

+

2 M X;9 = M X';o,lo = 2 M

As concerns the numerical value of B j for the I = 2' In practice this value might be reduced by 20

%

state of (Yb170)3f, by using the same data as above because of antishielding effects.

and :

Q, =

-

2.12 barns

_(p

_bis]) 6. Computation of the Mossbauer lineshape for a

transition (Of 2 9 .

-

We will only consider this case

<

J

11

a

11

J

>

= 2/63 ([5] p. 875) as the handling of other situations would be quite

we find that : similar.

For a transition (Of 2') it has been shown in ref. [8] BJ/h =

+

32.85 MHz. (12) that [I 1] :

r

withp = -

-

iw and : 2

1

<

(gv, 00) ($, Im) ( R ( (ZV', 00) ($p ', Im')

>

= 6,,,,

-

<

vp

I

R"I v'p'

>

16 TI,

<

(gv, 00) ($, Im) I X", (gv', 00) (&', Im')

>

= - 6 , ,

<

P' m'

I

XO

I

~m

>

=

-

a,..

(

pf m1

1 '

Jtt

+

1

_pin

)

(15)

4

where R" has been calculated in § 3, X' in

6

4 and X" labelling of the 80 x 80 matrix to the (different)

in Cj 5. labellings of

R"

on the one hand, and of X' and X" on

As already mentioned, the matrix to be inverted is the other hand. 80 x 80, but the computation time can be greatly

(6)

true spin 312, which corresponds to P = 312, Q = 112,

M = 1, N = 1, figure 1. In this case the relaxation supermatrix is given directly by eq. (34) of [8] and the Mossbauer spectrum can also be obtained by the

FIG. 1. - Relaxation spectra of a transition (0+ 2+) in the presence of coupling AJ I. J with a true spin J = 312 (Jeo = 0,

T = 0). Intensity Z(w) versus l i w / A ~ as a function of f i / A ~ Tls.

tehsor operator method of [4], which only requires the inversion of a 4 x 4 matrix. With respect to figure 2 of [I], one notices that the line on the right goes down much less rapidly than the others. This can be understood by inspection of the 4 x 4 matrix (Appen- dix). However the case J = 312 is not encountered in the rare earth group, where the most interesting cases are ([S] p. 858).

7

J = -+

r,

+

r,

+

r,,

(yb3+)

In the absence of specific experimental data relative to the last case (where the kets depend on x), we were

FIG. 2. - Relaxafion'spectra predicted for (Yb170)3+ in the

rs

state. A J / ~ = 300 MHz, B J / ~ =

+

32 MHz, r / 2 n = 69.4 MHz. Intensity Z(w) versus f i w / A ~ as a function of f i / A ~ Tls. The vertical bars in the top spectrum represent the positions and

intensities of the seven lines in the absence of relaxation.

content to compute the Mossbauer spectrum expected for the

r,

of (Yb170)3+ using the hyperfine parameters

- -

_-_{300 MHz,} BJ

_-

₌

₊

_{32 MHz}

h h (16)

and the values P = 112, Q = - 1116 ([5], p. 728), M =

-

2, N =

-

3. The result is represented in figure 2. It remains to find a metallic matrix in which

r,

would be the lowest electronic level of Yb3+, which requires that the Lea Leask and Wolf scheme be reversed, as it is the case for Dy3+ in Pd [13] or (3) Dy3+ and Er3+ in LaA1, and ErAl, [14]. However preliminary results on Yb JI?! seem to indicate a

r,

ground state (P. Imbert, private communication). On the contrary (Sm152)3 + should often

have a (well isolated) T, ground state and, in a metal, the mixing with higher J levels should be small, as crystalline field effects are reduced with respect to insu- lators. Unfortunately Mossbauer experiments on this isotope are difficult.

(3) This last example is just to show that the level scheme can

(7)

C6-84 C. MEYER, F. HARTMANN-BOUTRON AND D. SPANJAARD

Appendix, - For a true spin J = 312 and a transi- 63 1 tion (0+ 2'3, there are four hyperfine levels in the

+

₇

_~

_-

_{3 -}

TI,

_~

₎

excited nuclear state : F = 712, 512, 312, 112 and one

in the ground nuclear state : G = 312. The Mossbauer 4 f i 1

spectrum is given by eq. (45) (46)

(47)

of [4], where M 7 / 2 512 = M 5 / 2 712 =

- -

7 Tls

-

the matrix M corresponding to eq. (46) of [4] is 4 x 4.

As a function of Aj, Bj and

l/Tl,,

its only non zero 7 4 1

M 5 / 2 312 = M 3 / 2 512 =

- -

-

matrix elements, labelled by the values of F, are : 10

TI,

6 1

7

TI,

The spectrum of figure 1 has been computed with

M 5 i 2 5 , 2 =

Aj = 1, Bj = 0. The narrow line corresponds to the transition F =

712

-

G = 312 which is associated

18 1

M 3 / 2 3 / 2 =

-

(-

3 A J )

-

7 %

with matrix elements ( M 7 / 2 7 1 2 , M 5 / 2 7 1 2 , M 7 / 2 5 / 2 1

whose damping part is small.

References

[I] SIVARDI~RE, J., BLUME, M., Hyperfine Znt. 1 (1975) 283. [2] SCHERER, C., Nucl. Phys. A 157 (1970) 81.

[3] BLUME, M., Nucl. Phys. A 167 (1971) 81.

[4] CHOPIN, C., SPANJAARD, D., HARTMANN-BOUTRON, F., J. Physique Colloq. 37 (1976) C6-73.

[5] ABRAGAM, A., BLEANEY, B., Electron Paramagnetic Reso-

nance of Transition Ions (Clarendon Press, Oxford)

1970.

[5 bis] BARNES, R. G., MOSSBAUER, R. L., KANKELEIT, E., POINDEXTER, J. M., Phys. Rev. 136A (1964) 175.

LEA, K. R., LEASK, M. J. M., WOLF, W. P., J. Phys. Chem. Solids 23 (1962) 283.

[~~~HARTMANN-BOUTRON, F., SPANJAARD, D., J. Physique 36

(1975) 307.

[8] GONZALEZ-JJMENEZ, F., IMBERT, P., HARTMANN-BOU-

TRON, F., Phys. Rev. B 9 (1974) 95-Erratum, Phys. Rev.

B 10 (1974) 2134.

[9] GREENWOOD, N. N., GIBB, T. C., MaSSbauer Spectroscopy (Chapman and Hall, London) 1970.

[9 bis] Value available in the literature before the Conference. The most recent value is

-

2.14 barn, see k w s , G. M., WAGNER, F. E., POTZEL, W., communication to this conference. The value indicated in ref. [9], Q = 1 barn, was only tentative.

[lo] GONZALEZ-JIMENEZ, F., IMBERT, P., Solidstate Commun. I1 (1972) 861.

[ l l ] Taking account of Appendix I1 of ref. [7].

[12] SHENOY, G. K., DUNLAP, B. D., Phys. Rev. B 13 (1976) 1353.

[13] DEVINE, R. A. B., Solid State Commun. 13 (1973) 935. [14] DEVINE, R. A. B., POIRIER, M., CYRT, J. Phys. F. Metal