RELAXATION SUPERMATRICES AND RELAXATION MODELS IN THE INTERPRETATION OF MOSSBAUER PARAMAGNETIC RELAXATION SPECTRA

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RELAXATION SUPERMATRICES AND

RELAXATION MODELS IN THE INTERPRETATION

OF MOSSBAUER PARAMAGNETIC RELAXATION

SPECTRA

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

To cite this version:

C. Chopin, D. Spanjaard, F. Hartmann-Boutron. RELAXATION SUPERMATRICES AND

RELAXATION SUPERMATRICES AND RELAXATION MODELS

IN THE INTERPRETATION

OF MOSSBAUER PARAMAGNETIC RELAXATION SPECTRA

C. CHOPIN, D. SPANJAARD Laboratoire de Physique des Solides Universite Paris-Sud, 91405 Orsay, France

and

F. HARTMANN-BOUTRON Laboratoire de Spectrometrie Physique U. S. M. G., BP 53, 38041 Grenoble Cedex, France

Résumé. — Nous présentons ici quelques résultats récents relatifs au traitement de la relaxa-tion par des méthodes de perturbarelaxa-tion : 1) Il y a deux supermatrices de relaxarelaxa-tion R et S qui sont transposées l'une de l'autre ; il leur correspond deux superopérateurs d'évolution ; 2) La relaxation sphérique d'un spin S est décrite par 2 S constantes lt> ; le modèle stochastique de Scherer-Blume revient à prendre toutes ces constantes égales ; 3) Quand la relaxation est sphérique et que S = i le modèle de Scherer-Blume et le traitement de perturbation sont équivalents. Le spectre Môssbauer est alors donné par une relation simple déduite de la théorie stochastique ; quand 5 ^ 1 cette équivalence n'existe plus et il faut utiliser des méthodes tensorielles, qui abaissent considérablement les dimensions du problème. Les applications de ces résultats sont examinées.

Abstract. •— We report here some recent results relative to perturbation treatments of

relaxa-tion [1]. First, there are two relaxarelaxa-tion supermatrices R and S which are transposirelaxa-tions of one another, and two related evolution superoperators. Second, the spherical relaxation of a spin S is described by 2 S constants A* ; the Scherer-Blume stochastic model (or « RPA ») amounts to taking all these constants equal. Third, when relaxation is spherical and S — i, the Scherer-Blume model and perturbation approach are equivalent. This makes it possible to obtain the Mossbauer spectrum from a simple relation derived from stochastic theory. On the contrary when S # i there is no equivalence and one has to resort to tensor operator methods which considerably reduce the dimensionality of the problem. Applications of these results are discussed.

1. Relaxation supermatrices [2]. — It has been

shown previously that the effect of spin-lattice relaxa-tion of a Mossbauer atom is to add to its Liouville hyperfine hamiltonian 3to a dissipative part in the form of a relaxation supermatrix.

Care must be exercised however, as there are actually not one but two distinct relaxation supermatrices R and S. R characterizes the relaxation of an atomic operator Q in the Heisenberg representation, while S is relative to the relaxation of the atomic density matrix a in the Schrodinger representation :

£-(-i*{

+ S

) . . (2,

Similarly, there are two evolution superoperators, °\Ht) (introduced by Blume [3]) and ^(t) :

fi(0 s ^L(t) Q(0) = exp [ ( l K + R) t] 6(0) (4)

o(t) = TJ«) o(0) = exp [ ( - 1 3 ^ + s ) t] <T(0) . (5)

The relation between these two can be obtained by considering their expressions in terms of a lattice average :

Q(t) = ^ ( 0 6(0) = TraceIatt { U+(t) QU(t) pB } (6) a{t) = n5(0 <x(0) = Tracelatt { U(t) a(0) pB U+(t) } (7) where U{t) is the evolution operator of the total system (atom + lattice) and pB the Boltzmann density matrix of the lattice. With respect to eigenstates a, b... of the atom and r, r' of the lattice :

< ad |

nl(0

| bc> =

= Z<ar\ U+(t) \br'>< cr' \ U(f) \ dr > (prr)Boltz

rr' (8) < be | nj(0 | ad > = = YJ<br'\ Uit) | ar > (p„)BoItz < dr | U+(t) \cr'>. rr' (9) Then taking account of the relation :

< br' | U{t) | ar > = ( < ar \ U+(t) \ br' > ) * (10)

C6-74 C. CHOPIN, D. SPANJAARD AND F. HARTMANN-BOUTRON

we arrive at :

<

ad

I

%(t)

I

bc

>

=

(<

bc

I

w(t)

I

ad

>)*

= < c b I W ( t ) I d a > (11) from which it follows that :

R = ST (12)

i. e. R is the transposed Liouville matrix of S. At high temperatures and for simple relaxation models, R = S ;

but at low temperatures this is no longer true. The expression of S corresponding to a relaxation hamiltonian :

X1

=

Kq

Fq

(1 3)

4

(Kq : atomic operator, Fq : lattice operator), and with respect to eigenstates of X,, has been given in ref. [2] eq. (51) :

Generally speaking, relaxation effects can be intro- duced into any problem by replacing the evolution superoperator of the atom in the absence of relaxation :

v0(t) = exp

(-

f

~e; t) (15)

by the superoperator in the presence of relaxa- tion :

-

V(t) = exp

[ (-

x,'

+

S) t]

.

(16) However some difficulties arise when the atomic observable of interest appears as a multiple time inte- gral over a product of more than two usuaI evoIution operators U, (case of emission and scattering). A rule for handling such situations (by time ordering and breaking of averages) has been given in Appendix I of [2].

As an illustration let us consider the case of a Moss- bauer emitter. We have shown in ref. [2, 41 that its lineshape is given by :

z(o)

x

5:

exp

(

[io

-

i)

t] x

x Trace

{

TL(0)

.

T t ( t ) a,(l/T)

}

(17) where TL is the nuclear multipole operator responsible for the y transition and oI(l/T) is the average density matrix of the excited Mossbauer state I at the time of y emission.

Let us denote the excited Mossbauer electronuclear states by

I f

>

and the ground electronuclear states by

I

g

>.

By applying the rules of Appendix I of [2] we find that within the Liouville formalism :

with :

where ofn is the initid density matrix of state Ijust after feeding by radioactive decay of an upper state.

At high temperatures and in the absence of electronic rearrangement effects, o!,, and aI(l/T') are proportio- nal to the unit matrix. Then ofsfl(l/T) = dfl f 3 and by using eq. (1 1) we get the now well known result :

2. Description of electronic relaxation [5].

-

The dimensions of the relaxation supermatrices R and S are often very large and this both creates some computa- tional problems and introduces a great number of relaxation parameters. It is thus important to find situations where the number of parameters is very small and where the relaxation supermatrix factorizes. Such a case is provided by relaxation by the conduction elec-

trons considered in ref. [6, 71, which at finite tempera- tures (k, T % X,) has spherical symmetry. This means first that it satisfies the white spectrum approximation and second that the instantaneous fluctuations of the thermal bath are isotropic (notice that a necessary condition for spherical relaxation is that

k,

T 9 X, :

high temperature).

describing the spherical relaxation of an electronic spin S. This has been done in ref. [5] which is devoted to P.

A.

C . It is assumed that the electronic relaxation hamiltonian X I has the form :

where

Rf'

(p =

-

I,

...,

+

1) is the component of a tensor operator of order I adapted to S (I

<

2 S). One

then performs a multipole expansion of the electronic density matrix or :

a' =

C

Tk4(S) C T ~

kq

(22) (k

<

2 S, q =

-

k,

...,

+

k). The final result of the

relaxation calculation is :

with :

where W(SlkS

I

SS) is a Racah coefficient and

<

S

11

R,

11

S

>

a reduced matrix element. In other terms the most general spherical relaxation of an electronic spin S is described by 2

S

independent para- meters A,. When the tensor order of the relaxation is given, these parameters are no longer independent : as,

or by the conduction electrons (I = 1) :

By comparison, the Scherer-Blume relaxation model, or RPA [8, 91 which was used in recent stochastic theories [lo], amounts to taking

Lk

= A whatever k .

There is a priori no reason why this model should correspond to real physical phenomena. The only exception is the case S =

3,

where in the perturbation approach there is only one parameter A, = l/Tls. Then if we take

L

= 1/T1, the stochastic Scherer-Blume model and the perturbation theory are equivalent. As a final remark, we have already noted elsewhere that the approximation of spherical relaxation is not always valid, especially for phonons.

3. Factorization of supermatrices in the case of spherical relaxation.

-

We assume that the nuclear and electronic spins are coupled by an isotropic (free- atom type) hyperfine hamiltonian, leading to hyperfine states F (F = I

+

S) in the excited Mossbauer state

and to hyperfine states G (G = I,

+

S) in the ground

Mossbauer state. Also we assume a spherical relaxation of S. We shall see that when S =

+,

by virtue of the equivalence of Scherer-Blume and perturbation models in this case, the simplest way to obtain the Mossbauer spectrum is to use the Scherer-Blume model. On the contrary, when S #

3

it is necessary to resort to tensor operator methods.

3.1 ADAPTATION OF THE SCHERER-BLUME MODEL. - The Mossbauer lineshape at finite temperature is :

If we use normed tensor operators TL the trace is quite reeniscent of a perturbation factor in P. A. C . Then [ l l ]

m

I ( o ) a Re

I.

exp [(iw

-

3

t]

where &) is the Laplace transform of G(t).

On the other hand Dattagupta and Blume [I 11 have demonstrated that fo_r certain stochastic relaxacon pro- cesses, with stochastic frequency 1, there exists a simple relation ship between G in the presence of, and Go in the absence of relaxation :

It can be shown that this relation is also valid for the Scherer-Blume relaxation model.This is done by adapting to this problem a matricial demonstration already used for P. A. C . in Appendix I of 151.

Let us define in a decoupled basis :

we have that :

C6-76 C. CHOPIN, D. SPANJAARD AND F. HARTMANN-BOUTRON where :

S S ~ , O

= 1 1 4 2 s

+

1 . In terms of U$ : ~ t f ( t ) = Trace

{

U$+(t) u ~ ( o )

.

I S I ,

At that point the demonstration of [5] applies without any change. Notice that in [ 5 ] we assumed for simplicity an hyperfine hamiltonian

X,

= AI.S, but that the original demonstration of Chopin 1121 applies to any isotropic interaction.

When S =

3,

using eq. (28) and the relation :

the Mossbauer spectrum is obtained immediately if the hyperfine levels are known. As an example [I], in the case of YbAu in zero field this leads to an expression much simpler than that of ref. [7] _-- :

3.2 TENSOR OPERATOR METHODS. - Here too we simply adapt to the Mossbauer case a technique already used in P. A. C. [5, 121.

In eq. (26) the tensor operators

TP

are pure nuclear operators. In the presence of hyperfine structure let us introduce tensor operators adapted to the coupled system (I, S ) (I,, S ) :

""

vf

=

C

(- l)G+g

<

FfGg

I

K Q

>

I

F , f

> <

G,

-

g

1

.

f9

It can be shown that :

I I .

TLM

=

C

FG FG bL (

v - 1

FG with Then

KG'

bLFG

' ( o ) a Re

x

( 2 +

1,

exp [ ( i w

-

f

)

t ] Trace

{

'."

vFt(t)

FG v?(O)

M

F G F'G'

Let us now perform (I) an expansion of the off diagonal part (I, I,) of the density matrix o of the atom in terms of the V :

It can be shown that in the presence of the hyperfine interaction and of a spherical relaxation of S, the equation of evolution of ,,o takes the form :

d

- ("'"' &*) =

-

iwFrG, (F'G'

gg>

-

d t FG

x

PL(FG ++ F' G') (FG og*) (40)

where

Pk

is a relaxation coefficient to be discussed below. The solution of this system has the form :

It is then possible to demonstrate that as a function of the Laplace t r a n ~ f o r m Y ( ~ ) off (t) :

I(o) K (2 L

+

1) FG -, F' G', p =

2 F'G'

Notice in passing that this equation may also be written in a form similar to eq. (20) :

It follows from these results that for given multipole order L of the nuclear transition the computation of the Mossbauer lineshape requires the inversion of a single matrix whose dimension is equal to the number of couples (F, G ) , i. e. :

I(U) K ( 2 ~ + 1)

z

b : G ' ( ~ , ~ ' ~ f

FG

F'G'

[Inf (2 I, 2 S)

+

11 x [Inf (2 I,, 2 S )

+

I]

.

In order to get this matrix, it is necessary to compute the damping coefficients of eq. (40), which are related to the damping coefficients A L P of the electronic spin alone, eq. (24), by an expression similar to eq. (38 bis)

of [5] involving 6 j coefficients :

-

1 Jl

I <

S I I R l l I S > ) 2 B ~ ++ GF' G') =

z

-

[6,#

_{~ G G *}

+

t h 2 ( 2 S + 1 ) L, FG) biG

.

₍₄₃₎

These expressions may seem complicated, but in practice they are not difficult to handle, as tables of 6 j

symbols are available. As a first example, expression (34) for the Mossbauer lineshape of YbAu was originally obtained by this technique (which only required the inversion of a 2 x 2 matrix) ([5]xppen- dix 11, [12]). As a second example, in the course of the calculation relative to the

r8

quadruplet in [13], we have used this method to

.

compute the Mossbauer spectrum of a transition 0+ 2+ (L = 2) in the pre- sence of hyperfine coupling to a true spin S = $ relaxed by a fluctuating field ( I = 1). When I, = 0, we have that L = I, G = S and eq. (43) (40) (44) are replaced by :

( L , F ~ G /

-6~:

+ S ~ L , F G ) =

-

i w F , ~ - P;(F' G o FG) (46) I(@) a

c

(- I ) ~ + ~ ~ 4 2 F

+

1) (2 F'

+

I) (L, F' G

FF'

and (assuming 1 = 1) :

With S =

9

and I = 2, the spectrum is obtained by the inversion of a single 4

x

4 matrix, thus giving a simple check for the complex program relative to the general T 8 case, which involves a 80 x 80 matrix.

- 1

i

- p - $ X o " + S

Applicability of these results : it is restricted by the condition that the hyperfine hamiltonian be a free- atom like hamiltonian. In cubic symmetry this is true for

T,

or

r,

which are equivalent to S = -$ and have no quadrupolar effects. On the contrary,

r,

and

r,

are equivalent to S = 1 only as concerns the magnetic part ([14]) p. 740). It is well known that T 8 is not Cquivalent to S = $ ([14] p. 721). Another possibility would be that of (( S )) state ions with a true spin S (Fe3+, Gd3+), but in this case there often exists a spin hamil-

tonian which is comparable to the (small) hyperfine structure ([I41 p. 440, 335), and should be included in the calculation. It is possible that the main interest of the tensor method will be to provide a simple check of the long programs needed by more refined calculations.

C6-78 C . CHOPIN. D. SPANJAARD AND F. HARTMANN-BOUTRON

References

[I] An elementary introduction to the problems considered in [71 GONZALEZ-JIMENEZ, F., IMBERT, P., HARTMANN-BOU- this paper can be found in HARTMANN-BOUTRON, F., TRON, F., Phys. Rev. B 9 (1974) 95. Erratum Phys. Rev.

Ann. de Phys. 9 (1975) 285-356. B 10 (1974) 2134.

[2] HARTMANN-BOUTRON, F., SPANJAARD, D., J. Physique 36 f8] S C H E ~ ~ ~ y C.7 Nuel. Phys. A 157 81.

(1975) 307. [9] BLUME, M., NUCI. Phys. A 167 (1971) 81.

[3] BLUME, M., Phys. Rev. 174 (1968) 351. [lo] SIVARDIERE, _{[ I l l DATTAGUPTA,} J., _{S., BLUME,} BLUME, M., Hyp. Int. 1 (1975) 283. _{M., Phys. Rev. B 10 (1974) 4540.} [41 HARTMANN-BOUTRON, F., SPANJAAR~, D., J . 33 _{[12] CHOP^, C., TheSe de 3e cycle, Orsay 1975.}

(1972) 285. _{1131 MEYER, C., HARTMANN-BOUTRON, F., SPANJAARD, D.,}

[5] CHOPIN, C., SPANJAARD, D., HARTMANN-BOUTRON, F., J. J. Physique Colloq. 37 (1976) C 6-79.