Paramagnetic relaxation effects in perturbed angular correlations for arbitrary electronic relaxation time

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Paramagnetic relaxation effects in perturbed angular correlations for arbitrary electronic relaxation time

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

To cite this version:

C. Chopin, D. Spanjaard, F. Hartmann-Boutron. Paramagnetic relaxation effects in perturbed angular correlations for arbitrary electronic relaxation time. Journal de Physique, 1975, 36 (10), pp.961-978.

�10.1051/jphys:019750036010096100�. �jpa-00208341�

PARAMAGNETIC RELAXATION EFFECTS IN PERTURBED ANGULAR CORRELATIONS FOR ARBITRARY ELECTRONIC RELAXATION TIME

C. CHOPIN, ^D. SPANJAARD, ^F. HARTMANN-BOUTRON

Laboratoire de Physique des Solides (*), Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay, ^France (Reçu ^{le 5 mars 1975,} accepté ^{le 9 mai} 1975)

Résumé. ^- Les traitements de perturbation ^utilisés jusqu’à présent pour calculer les effets de la relaxation paramagnétique ^{dans les} expériences de corrélations angulaires perturbées ^{yy n’étaient}

valables que pour des temps de relaxation électronique ^{très courts. Nous tournons cette difficulté}

en adaptant ^{à ce} problème la nouvelle théorie de perturbations ^{mise au} point récemment par Hirst et d’autres auteurs afin d’étudier les effets de relaxation dans les spectres Môssbauer. Moyennant l’hypothèse de relaxation électronique sphérique, il devient alors possible de calculer les facteurs de

perturbation ^en fonction de paramètres de relaxation qui ^sont directement reliés au hamiltonien de relaxation microscopique. Nous comparons ^nos résultats à ceux de la théorie stochastique ^{de Scherer}

et Blume.

Abstract (**). - Previous perturbation treatments of paramagnetic relaxation effects in _yy PAC

were limited to the case of very short electronic relaxation times. We circumvent this limitation by invoking ^{a new} perturbation theory recently elaborated by ^{Hirst and} others for handling ^relaxation

effects in Môssbauer spectra. ^{Under the} assumption ^of spherical electronic relaxation we are then able to compute ^the perturbation ^{factors as} functions of certain relaxation parameters ^{which are} directly ^{related to the} microscopic relaxation hamiltonian. We compare ^our results to those of the stochastic theory of Scherer and Blume.

Classification

Physics ^Abstracts

4.200 ^- 8.610 ^- 8.680

1. Introduction (1). ^{- Previous} perturbation ^treat-

ments [1 -3] of paramagnetic relaxation effects in PAC

were limited to the case of very short electronic relaxa- tion times Tls. The aim of this _paper is to show that the relaxation theory ^devised by Hirst and others for Môssbauer spectra [4-6], ^{allows one to} compute ^the

perturbation ^factors Gkk (t) ^{for a} Tls ^of arbitrary ^size.

Let us consider a paramagnetic impurity ^{in a solid}

with a radioactive cascade I; I k _If and assume

that the intermediate state I observed in PAC exhibits

a hyperfine interaction : Xhf ^{= AI.} S, where S is the effective spin of the lowest electronic level of the impu- rity. Coupling of S with the lattice gives ^{rise to an}

electronic relaxation characterized by Tis.

The expressions ^{for the} perturbation factors in the

absence of relaxation are well known and correspond

to oscillating ^{behaviour at the} hyperfine frequencies [7].

On the contrary ^when Tis ^is very short, the relaxation

gives ^{rise to a} pure attenuation of the correlation which was computed by Abragam and Pound [1].

In order to obtain this result they assumed that the effect of relaxation is to confer to the hyperfine ^hamil-

tonian a fluctuating character :

This hamiltonian, whose correlation time Te ^is clearly equal ^to Tis, is then treated by ^a perturbation ^method

similar to that used in chapitre ^{8 of} Abragam’s

book [8]. The condition for the validity ^{of the} pertur- bation method is that « H12 > z2/2) ^{« 1. Conse-}

quently ^{the treatment of} Abragam and Pound is limited to very short relaxation times : (A/fi) Tis « ^1.

The intermediate case (ale) Tis - ^{1 was} recently investigated by ^Scherer [9] ^{and Blume} [10] ^{with the} help ^{of a} stochastic model in which the lattice, repre- sented by ^a stochastic hamiltonian Ri acting ^on S, induces transitions between the eigenstates ^{of the} coupled electronuclear system (I, S). ^{In this} approach

one has therefore a static hamiltonian : (*) Associé ^{au C.N.R.S.}

(**) A preliminary ^{account of the} present ^{work has} already ^been published ^{in C.} Chopin, ^D. Spanjaard, ^F. Hartmann-Boutron, Compt. ^{Rend. 278} (1974) ^{739. It was also the} object ^{of a short}

communication to the Uppsala Conference on Hyperfine ^Interac- tions (June 1974). ^{In the} figure caption ^{of both} papers ^one should read time unit : T ⁼ 2 h/5 ^A.

( ’ ) ^{In this} introduction, ^{we do not intend to} mention all previous

studies in this field, but only ^those papers directly ^{related to the} present work. Further references can be found in references [2, 3, S, 7, 9, 10, 11].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019750036010096100

and a fluctuating hamiltonian corresponding ^{to ran-}

dom independent pulses acting only ^{on S :}

As discussed by Clauser and Blume [11], ^the average effect of one of these pulses ^can be characterized by ^a

transition superoperator Tav (in the Liouville _represen-

tation) :

Scherer and Blume assume that the effect of each pulse

is to make the direction of the electronic spin ^{S random} and, ^{as mentioned} by ^Blume ([10] ^p. 83), ^{this is equi-}

valent to assuming that with respect ^to eigenstates ms, ms ^of Sz :

With these hypotheses ^the problem ^is clearly isotropic.

Hence :

On the other hand, using ^the explicit ^{form of} 7:av, it is

possible ^{to establish a} simple relationship ^between

the Laplace transforms Gk(p) ^and Gk0(p) ^{of the} perturbation ^factors Gk(t) ^and GI(t) ^{in the} presence

or in the absence of relaxation. The quantity being

the _average frequency ^{of the} pulses, ^{assumed to} obey

a Poisson statistics, Blume demonstrates that (eq. ⁸

of [10]) :

Gk(t) ^{is then} easily ^obtained by ^inverse Laplace ^trans-

formation of &(p). ^Numerical examples ^show that,

as expected, ^when À varies from zero to infinity, Gk(t) ^{goes from a} purely oscillating behaviour, ^to

a damped oscillating behaviour, ^{then to a} purely damped behaviour and becomes finally equal ^{to 1}

when ⁼ oo. This evolution is qualitatively very

satisfying. However the parameter ^{has no clear} relationship ^{with real} microscopic phenomena taking place ^{in a} solid, ^{which are due to} stationary spin-

lattice couplings ^{and not to random} pulses.

Finally ^a perturbation theory ^{has been} developped by Hirst and others [4-6] ^for handling ^relaxation

effects in Môssbauer spectra. ^{It relies on a} physical analysis which is somewhat similar to that of Scherer

Blume, i.e. there is also a main hamiltonian Xo ^{= AI. S}

and a relaxation hamiltonian H1 which describes the

coupling of S with the lattice, H1 ^{is a} microscopic stationary hamiltonian which can be written in explicit

form when the relaxation mechanism is known [12].

h1 is treated by ^the perturbation ^method (Abra-

gam [8] Chap. 8) ^but contrary ^{to the} Abragam-Pound theory, the correlation time 7:c associated with Xl

is now a lattice correlation time, ^{which is} always very short (10-13-10-15 s) ^so that the criterium :

is always satisfied. It then becomes possible ^{to treat the}

intermediate region A g T1S ’" 1, provided ^{that one}

AÏ ^{is p}

abandons the secular approximation ^{used in most}

relaxation theories. Hirst [4] considered a spin-lattice coupling ^{of the} form H1 ^{= - 2 JS.s} (relaxation by the conduction electrons). ^{In the} present paper ^we shall consider more general relaxation mechanisms.

On the other hand, ^{in order to} lower the dimensions of the differential systems ^{or matrices which come into}

’ play, ^{we shall} generally ^assume that the relaxation has

a spherical ^character [3] ; this makes it possible ^{to use}

tensor operator methods which have already proved

to be _very powerful ^{in such} problems [3], [13, 14].

As already mentioned, ^the perturbation approach

of Hirst has the advantage ^of being directly ^{related to}

the microscopic processes in a solid. In the case of the Môssbauer effect, it has made it possible ^{to extract}

very fundamental and interesting information from the low temperature ^Môssbauer spectra ^of Ybl’°Au [5], [15, 16]. Since PAC have wider applica- bility than the Môssbauer effect, ^one may hope ^that

the extension of Hirst’s theory ^{to this case} will enable

one to obtain similar information on a greater ^number of paramagnetic radioactive impurities.

2. Application of Hirst’s theory ^to PAC. General formulae. - In this section we first recall some

previous ^{results relative to the} relaxation of a bare nuclear spin under the effect of its coupling ^{with the} lattice, ^{and we} indicate how these results can be trans-

posed ^to the relaxation of a bare electronic spin. ^We

then consider the case of two coupled ^{nuclear and}

electronic spins ^{in the} presence of electronic relaxa- tion. We introduce various tensorial bases adapted

to this problem ^{and we} derive the corresponding general expressions ^{of the} perturbation ^{factors, which}

we relate to the temporal evolution of the density

matrix. Finally ^we establish the equation ^{of motion}

of the density matrix and we use it to obtain the fundamental expressions ^which will be used in ^sec- tions 3 and 4 to compute numerically ^the perturbation

factors both in the absence and in the _presence of an

external magnetic field. We also _compare our results with other theories.

2.1 SUMMARY OF PREVIOUS RESULTS. ^- In refe-

rence [3] ^we considered the relaxation of a nuclear

spin I coupled directly ^{to the lattice} by fluctuating

interactions having spherical symmetry. By ^{this we}

mean that 1) ^the spectral densities of the lattice corre-

lation functions which _appear in the relaxation _equa- tions do not depend ^{on the} frequency (white spectrum

approximation) ^{and that} 2) the instantaneous fluctua- tions of the lattice are isotropic. ^The assumption ^of

spherical symmetry requires ^{that the} temperature kB T ^be big compared ^{to the} hyperfine hamiltonian,

a condition which is always fulfilled in standard PAC

experiments ; ^on the other hand ^{one must be aware}

that some relaxation processes might ^not fulfill the conditions for spherical relaxation.

In the case where they ^are fulfilled, ^{it is} interesting ^to

use a multipole expansion ^of the density matrix of the nuclear spin :

with :

It then _appears that the relaxation equations ^{of the}

various a* ^are completely decoupled :

and that when relaxation is due to a tensorial interac- tion of order

1 je 1 = E p R p 1 (1) F ^with 1 2 1 )

the relaxation constants lik ⁼⁼ IlÀk ^are given ^{as func-}

tions of a unique parameter Jl by ([3] ^eq. (29))

where W is a Racah symbol ( = 6 j coefficient) ^{and :}

(the bar denotes an average ^over the lattice). ^{In the}

case when several interactions with different tensorial orders are acting ^on I, ^and assuming ^that they ^are statistically independent, ^{we have :}

On the other hand we have demonstrated that (13) :

where the G ^are the perturbation ^{factors which can be}

observed in a PAC experiment ^{on a cascade with I as}

its intermediate state. Consequently ^{the G can be} easily ^obtained by integration ^{of the} equation ^of

motion of the of ^{and in the} simple ^case eq. (9) ^we immediately get

In addition to eq. (13), ^{one can} also establish the

general ^relation [ 13] :

where A is the evolution operator ^{in state I.}

In the present paper ^we shall have to deal with a

much more complicated situation than the preceding

one, i.e, two coupled spins ¹ and S with relaxation

acting this time on S alone (2). ^{In order to} simplify

it we shall assume that, in the absence of nuclear spin,

the relaxation of S has spherical symmetry. ^{We can} thus transpose ^{to S} the results which were previously

derived for I ; i.e., ^{if we} perform ^a multipole expansion

of the density matrix of S as

then in the _presence of a spin ^lattice coupling

we have :

with

Notice that by definition of Tls , Al ⁼ 1 /T1s ^and

that if relaxation is due to a single tensorial interac- tion (given 1), ^all Âk’s ^are proportional ^to 1 / Ti s.

Let us also recall that the conditions for spherical symmetry ^of the electronic relaxation have been discussed elsewhere. They ^{are fulfilled} by ^standard

conduction electron relaxation in metals and also

by ^certain phonon relaxation _processes in high symmetry ^sites [17].

2. 2 GENERAL EXPRESSIONS OF THE PERTURBATION FACTOR IN THE PARAMAGNETIC CASE. ^- In the _presence of the degrees of freedom of the electronic spin ^the general expression ^{for the} perturbation ^{factor eq.} (15)

must be replaced by [18] :

where A now is the evolution operator ^{in the electro-} nuclear manifold (1, S). It is well suited for computa-

(2) ^In reality ^{1 is also} coupled directly ^{to the lattice but this} coupling ^is negligible compared ^to the electronic spin-lattice coupling.

tions using ^{a basis of} eigenstates ^of I2, S2, IZ, Sz (decoupled ^{basis ml} ms).

z

On the other hand, ⁱⁿ the absence of external _magne- tic field the eigenstates ^{of AI.S} are 1 F, Fz > ^which

are eigenstates ^of I2, S2, ^{F2 =} (I ⁺ S)’ ^and

(coupled ^basis FFz). ^{Tensorial operators adapted to}

this coupled ^{basis are} [19] :

with hermitian conjugate :

and orthogonality properties :

The Tkq(I) ^{are related to the} F’FV: by (Edmonds [20] eq. (5.4.1) ^and (7.1.7)) :

with :

Inserting ^these equations ^into (19) ^we get ^{a new} expression ^{for G}

or alternatively ^{in terms of} Heisenberg operators :

We now show how G can be deduced from the equation of evolution of the density ^{matrix Q of} the electronuclear manifold (IS) ^{in the} basis 1 FFz ). ^{Let us} perform ^a multipole expansion ^{of 6 :}

It is _easy to show that :

Consequently :

where the f are ^obtained by integrating ^the equation ^{of motion ofoB}

Notice that G could also be obtained directly by writing ^the equation ^of relaxation of V in the Heisenberg representation (while ^the equation of relaxation of u is written in the Schrôdinger representation). ^{In the} high temperature ^{case we are} considering here, ^{the two} relaxation Liouville supermatrices ^R and S associated to V and

u are exactly ^{the same} [6] and this second method using V(t) ^{does not} require any special ^care (this ^{would not be}

true at low temperatures, ^see [6]).

2.3 PERTURBATION FACTORS IN ZERO EXTERNAL FIELD. - The hamiltonian of the problem is :

The eigenstates ^of Jeo are 1 FFz > ^with eigen-energies :

Let us expand u ^{in terms} of FGV:. The contribution of Jeo ^to dcr/dt is trivial :

On the other hand we have shown in the preceding paragraph ^{that, in} the absence of nuclear spin, the relaxa- tion of S is simply described in terms of tensor operators Tl(S). When the nuclear spin is added this will remain true with respect ^{to direct} products Tk’q’(I) Tt(S). ^We shall therefore introduce a multipole expansion adapted ^to

a decoupled basis, ^{work out} the relaxation in this basis and _go back to the coupled ^basis.

Let us define :

The relations between the V and the U are the following [ 19]

When we expand ^{Q in terms of ’I’s} Uk ^{a trivial} adaptation ^of eq. (17) ^{leads to a} relaxation term

Transformation to u,’ ,’ ^is then achieved by using eq. (35) ^and (36) and, ^after defining :

we finally get ^the following equation of motion for 6 :

As already mentioned the perturbation ^{factors are obtained} by integrating ^this system, determining ^the

f[g, i §?g] ^(eq. (29)) ^{and inserting them into eq. (30).}

Notice that when Â.ks ^is given by eq. (18) the summation over kl, ks ⁱⁿ eq. (38) ^{can be} performed ^{with the} help ^{of an} identity ^{due to} Ducloy [38] ^and eq. (38) ^{becomes :}

Several comments can be made about _eq. (39). First,

in deriving ^{it we} have said that the _passage from _eq. (17)

to eq. (37) ^was trivial ; ^{this is true} only ^{as far as the}

white spectrum approximation ^is valid, ^{but this}

condition is precisely realized here since we assume a spherical relaxation of S. Second the damping

coefficients Pk depend on rall Àks’ ^{When S = 1/2 there}

is only ^one Àks : ^{Àl -} 1/Tis. ^For general S ^{there are}

2 S of them : Àl, À2, ^..., À2s ^{which can} be considered

as independent parameters_ ^{unless we consider a} specific relaxation mechanism (given 1) ^{in which}

case, by ^virtue of eq. (18) they ^{can all be} expressed ^as

functions of Ài ⁼ 1/Tis. ^{Third in eq.} (39) ^{we have} coupling only between . s with the same k and _q, and for given ^{k the} equation ^is independent ^of q.

As will _appear later, by comparison ^{with the} magnetic

field term which is uniaxial, this feature arises from the fact that Âqks =- îks ^is independent ^of qs. A conse-

quence of it is that (see ^eq. (29) ^and (30)) :

Let us consider a specific example : I = 2, ^{S =} 1/2

which leads to two hyperfine ^{levels F =} 5/2 ^and

F ⁼ 3/2. Taking ^{account of eq.} (40), ^we only ^{have to}

compute G22 (t) ^and G22(t). ^On the other hand for

given k and q the number of coupled eq. (39) ^is equal ^to

the number of couples (F, G), ^i.e. [Inf (2 ^{1, 2} S)+ 1]2,

which here is equal ^{to 4. In} practice ^we have 4 equa- tions when k ⁼ 2 and only ^{3 when k = 4} (since, by

virtue of the triangle ^rule 3/2,3/2 V: = 0). ^Conse- quently ^{in order to obtain} G22 ^and G0044 ^{we have to}

solve respectively ^a system of four linear differential

equations ^{and a} system of three linear differential

equations.

For comparison ^{let us} consider what the dimensions of the problem would be if we did not use tensorial

operators. In ^terms of the Liouville evolution _super- operator cU(t) ^{or reference} [6], eq. (19) gives :

in which 9Y(t) describes the evolution of the density

matrix u :

In the _presence of relaxation cU(t) ^{has the} general

form [6] :

where Hô is the Liouville hamiltonian associated with

ico, and R describes the effects of relaxation. The dimensions of the corresponding Liouville matrices

arme [(21+ 1)’(2S + 1)2] ^x [(2 1 + 1)2 (2 S + 1)2]

i.e. 100 ^x 100. In order words ’tJ(t) would be obtained

by solving ^a system ^{of 100} coupled differential _equa-

tions. The simplification introduced by ^{the tensor} operators ^is clearly considerable.

However even with this simplification the numerical solution of system (39) ^{and the} subsequent computa- tion of the perturbation ^factor generally require ^a computer. ^This will be done in section 3. Before

undertaking ^{it we} may already ^{check our} results in

some limiting ^{cases and} compare ^our relaxation model with that of Scherer and Blume.

2.4 COMPARISON WITH OTHER THEORIES. ^- First in the absence of relaxation, solution of system (39) ^is

trivial and with help ^{of eq.} (29) (30) ^we immediately

get the standard expression ^{for the} perturbation

factor in this case ([7] ^eq. (96), [18]) :

Second, in the introduction we have mentioned that the relaxation model of Scherer and Blume for elec- tronic relaxation amounts to taking ^{a transition} operator :

The corresponding ^Liouville relaxation operator

is [11] :

(2/2 Tt ⁼ frequency ^{of the} pulses).

If we insert this expression ^{of R into the} equation

of motion of the density ^{matrix 6’ of} the bare elec- tronic spin (I = 0) :

and if we perform ^a multipole expansion ^{of (1’} (eq. (16))

we easily ^{arrive at} the relaxation equations :

In other words the Scherer and Blume model amounts to assuming ^{that all} multipole components ^{of the} electronic spin ^{relax with the same time constant :}

whatever

One must now ask two questions : first, ^{can this}

condition be satisfied by microscopic processes in ^a

solid; second, ^{does it} correspond ^{to a} situation of

physical ^{interest ?}

We have already ^seen that for all spherical ^relaxation

processes Âk’,s ⁼ Àks’ ^{Then eq. (50) reduces to the}

system :

There are 2 S values of ks and therefore _{2 S equa-} tions. On the other hand we have shown that (eq. (18)) :

with 1 , 2 S.

Inserting (52) ^into (51) ^{we see that we have to solve}

a system ^{of 2 S} equations with 2 S unknowns which

are the quantities

depending ^{whether we use} normalized tensor operators

or not). ^{Such a} system always ^{has a} unique ^solution.

Therefore it is always possible ^{to find one} combination of microscopic processes (and ^one only) for which the

theory ^of Scherer-Blume applies.

On the other hand there is no reason a priori why

this precise combination would be of physical ^interest.

There is - however an exception which is the simple

case S ⁼ 1/2. ^{Then in our} model there is only ^one Àks corresponding ^to ks ^{= 1 :} Àl = I/Tls; in ^{that case}

if we make the identification +-+ I/T1s, ^the perturba-

tion factors computed by ^our theory ^or by ^Scherer-

Blume’s theory should be identical. On the contrary if S = 1 and if we assume for example that the relaxa-

tion of S is due to a coupling ^{with the conduction}

electrons (1 = 1), we ^{have that} À,1 ⁼ 1 / Tis ^and

which is not compatible ^with eq. (50) ; ^{we therefore} expect that the results obtained by ^{the two theories}

should be different.

Finally, in the limit of _very fast relaxation Â, » Alh

we must check that our theory ^{reduces to} Abragam

and Pound’s result [1] :

with

The relevant demonstration, ^{as well as a different} comparison with Scherer and Blume, ^are given ^{in the} Appendix ^1.

2. 5 PERTURBATION FACTOR IN THE PRESENCE OF A MAGNETIC FIELD. ^- In this case we must add to

Jeo the electronic Zeeman coupling gJ.1-B Sz Ho, ^{and in}

the coupled ^{basis we have to} compute commutators of the type :

This is achieved by going ^{from the} coupled ^{basis to the} decoupled basis and back with the help ^of eq. (34), (35), (36). ^{We are then led to introduce a} coupling

coefficient :

In terms of _u, the equation ^{of motion of J is :}

It appears that now the only good quantum ^number of the problem ^is q, in other words we have couplings

between J£§§ ^with different k ; ^moreover the coefficient

uFG kF’G’ ^{depends on . The} perturbation factors of interest in PAC will be all the GZZ, ^with k, ^{k’ even.}

As an example ^{let us assume that 1 = 2, S =} 1/2.

In a perpendicular geometry (ki, k2 ..L Ho) ^{we must} compute ÉÎÎ> rT22, G24, G24 G24 G44 G44 G44 G 44 and similar terms obtained by reversing ^{the order}

of k or the sign ^of q. The dimensions of the differential systems associated with the different q’s ^{are : 18 for} q = 0 (16 forq = ± 1), 12 forq = ± 2,(8 for q ±3),

4 for q ⁼ ± ⁴ (1 for q ⁼ ± 5).

It appears that the use of the coupled basis and of tensor operators has little interest in this _case, this

being clearly ^{due to} the fact that the external field lowers the symmetry ^from spherical ^{to uniaxial. In}

addition this basis is not adapted ^{to the} high ^field