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Study of radioactive impurities in solids. - II. Effects of relaxation and radiofrequency
D. Spanjaard, F. Hartmann-Boutron
To cite this version:
D. Spanjaard, F. Hartmann-Boutron. Study of radioactive impurities in solids. - II. Ef- fects of relaxation and radiofrequency. Journal de Physique, 1972, 33 (5-6), pp.565-577.
�10.1051/jphys:01972003305-6056500�. �jpa-00207282�
STUDY OF RADIOACTIVE IMPURITIES IN SOLIDS.
II. EFFECTS OF RELAXATION AND RADIOFREQUENCY
D. SPANJAARD (*) and F. HARTMANN-BOUTRON
Laboratoire de Physique des Solides (**), Faculté des Sciences, 91-Orsay (Reçu le 3 janvier 1972)
Résumé. 2014 Les caractéristiques du rayonnement nucléaire émis par une impureté radioactive dans un solide, dépendent des propriétés des niveaux mis en jeu par les processus radiatifs. Dans le présent article nous étudions comment l’évolution de ces niveaux est affectée par les phénomènes
de relaxation et par l’application d’un champ de radiofréquence. Les résultats obtenus sont appliqués
aux expériences de corrélations angulaires perturbées et d’orientation nucléaire (calcul des facteurs de perturbation, effet de la saturation radiofréquence dans les expériences combinant la résonance
magnétique et l’orientation nucléaire).
Abstract. 2014 The nuclear radiation characteristics of a radioactive impurity embedded in a solid
matrix, depend on the properties of the levels involved in the radiative processes. In this paper
we investigate the evolution of these levels, in the presence of relaxation phenomena and of a radio- frequency field. The results which we derive, are then applied to perturbed angular correlations and nuclear orientation experiments (calculation of the perturbation factors, effect of radiofrequency
saturation in NO-NMR experiments).
Classification Physics Abstracts 12.10201318.00201318.20201318.54
1. Introduction. - In the first part of this paper [1]
we have seen that the angular distribution of the second y emitted in a y 1-y2 cascade is essentially
controlled by the evolution of the nucleus in the inter- mediate state I. We shall now investigate this evolution
when the interactions seen by the nucleus are repre- sented by a hamiltonian :
hw. I. represents the effect of the external magnetic
field plus (eventually) an average hyperfine field ; while Je1 (t) is associated with the fluctuations of the
hyperfine interactions around their mean value
(Xl = 0). These fluctuations are responsible for the
relaxation processes (1) and are characterized by some
correlation time Finally JeRF is the radiofrequency
hamiltonian associated with a circularly ipolarized
RF field.
In the present study we shall first recall the results previously obtained by us and other authors as concerns the effects of relaxation alone (NRF = 0) on perturbed angular correlation and nuclear orientation
experiments. We shall review the basic assumptions underlying the different theories and discuss their (*) This work is based on a part of the thesis of Daniel Spanjaard (ref. [14]).
(**) Associé au C. N. R. S.
(1) In all this paper we shall assume that the relaxations of the very dilute radioactive nuclei are decoupled, i. e. that there
is nothing comparable to a « spin temperature ». (See ref. [14]
p. 86).
range of validity and their applicability. In the second
part, we shall consider the effect of an RF field on
both types of experiments, either in the absence or
the presence of relaxation.
II. Relaxation effects in perturbed angular correla-
tion experiments. --- 1) GENERAL CONSIDERATIONS. -
A first study of relaxation effects in P. A. C. has been
published in ref. [2] and [3]. The problem consists in
calculating the perturbation factor Gkik2(t, t’) in the
presence of relaxation acting on level I. Two methods
were used :
a) For « short » ’Lc, a perturbation treatment which
enabled us to generalize the Abragam-Pound theory.
b) For « long » Te? the stochastic method of Ander-
son and Weiss, which has the advantage of being
valid whatever ’Lc, but, despite recent improvements [4]
only allows the investigation of approximate models.
In ref. [3], we have considered the rather trivial case
of a longitudinal fluctuating field.
As concerns case a) i. e. for « short » it has been
found that in two particular cases, the relaxations of the different Qk were decoupled, thus leading to the
very simple form (2) :
(2) Let us mention that when the static hamiltonian Ko contains a quadrupolar term, one never gets such simple results
since then the evolutions of the ak’ due to Jeo are not decoupled.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01972003305-6056500
These two cases were :
- « Spherical relaxation », i. e. instantaneous fluctuations with spherical symmetry, and co.,r,, « 1.
Then Â’ = Âk was given by eq. (29) of ref. [3].
- Relaxation through a fluctuating field. Then
(eq. (38) of [3]) :
These results have been derived under the assumption
of a « hi g h » température (kBT « kBT 1). , Indeed, as
shown again in [1], only under this condition, are the usual formulae for P. A. C. valid.
Formulae identical to ours have been obtained inde-
pendently by H. Gabriel [5] with the help of a diffe-
rent method, the principle of which will be indicated later.
In our own work, we have made use of the standard
theory of relaxation, as developed, for example, in chap. VIII of the book by Abragam [6]. This particular theory is only valid when the « observation time » is
long compared with the correlation time Te’ This condition is always fulfilled in NMR experiments.
But, as was first noticed by Gabriel, it may happen
that it is not satisfied in some P. A. C. experiments,
in which the observation timern (lifetime of level 1) (3)
turns out to be shorter than Te’ In order to deal with this situation Gabriel has extended his previous theory
of relaxation along the same lines [7] [8]. But, as we shall see below, this new theory is in fact valid under somewhat limited conditions. Before looking at these questions in more details, let us mention briefly that, contrary to all these « relaxation type » theories, the stochastic theories have the superiority of being valid
whatever Te and in.
Let us now come back to the relaxation type theories.
We shall first recall that the physical quantity of inte-
rest in integral P. A. C. experiments is a linear combi- nation of the 9", (:J 03C4n(1/03C4n)defined by :
In the simple cases where the relaxations of the Uek
are decoupled we have found that Gzf is given by an expression of type (2). Then :
This expression clearly shows that in order to be able to observe the integral P. A. C., one must have
COn in N l. On the other hand, in differential P. A. C.
it is sufficient that Ú)n in > 1.
(3) !"n = 1/l’2 in the notations of rsf. [1].
Let us now recall the main steps of the calculation of Gkk,(t, t’). In ref. [1] we have demonstrated the fol-
lowing general relationship between the density
matrix ou of level 1 at two different times (eq. (53) of
ref. [1]) :
The Gkg,"’ k can therefore be obtained by integration of
the equation of evolution of the density matrix under the effect of the hamiltonian Je = Jeo + XI(t). In
the interaction representation (u -+- p, Je1 -+- £,), this
equation can be written after iteration) :
Eq. (7) is exact. In the classical theories of relaxa-
tion [6] (which are valid when Ii -2 1 z 1), several approximations are made on eq. (7) ; they mainly
amount to :
- averaging independently over JC, and p,
- replacing p(t") by p(t) in the integral,
- extending the lower limit of the integral over t"
to - oo. Eq. (7) then reduces to a simple differential system ((6), chap. VIII, eq. (34) and (35) ; and Slich-
ter [9] p. 148, eq. (35)).
The starting point of Gabriel’s new theory [7]
consists in keeping only the first one of the three above approximations. Then eq. (7) transforms to an integral equation, the form of which can be consi-
derably simplified by resorting to the Liouville forma- lism [10], and which is then solved by Laplace trans-
formation. Only the second order correlation functions
of Je1 (t) appear in the final results of the calculation ; this is not surprizing in view of the approximation
retained by Gabriel, but suggests that this theory is
in reality a perturbation theory with limited validity.
The relevant conditions of validity are obtained
most easily within Liouville formalism. Let Lo, Ll, LR
be the Liouville operators associated with Jeo, JC,
and the lattice hamiltonian JCR. In the normal repre- sentation, it is found that (eq. (29) and (35) of [5]) :
with (eq. (30) of [5]) :
in which M(t) is given by the exact expression (eq. (19) of[5]):
Gabriel’s approximation consists in neglecting the Li term in the exponential factor of eq. (10). M(t) then
becomes a second order operator with respect to 3C,,
in agreement with our previous remarks. In order that this approximation be legitimate, it is necessary that, during the physical « observation time », LI t be very small compared to 1. According to eq. (9), the « obser-
vation time » is of order in. The theory of ref. [7] will
then be valid only if
Let us now investigate how this restriction limits the
physical effects predicted by this theory. In a very
simple case, it has been found that (eq. (4) of ref. [7])
instead of the classical result (,r,, « Ln)
It is seen that when ’C,,/rn » 1, T,, is replaced by zn in the relaxation term. There is a simple physical expla-
nation for this result : when the lifetime in of the intermediate nuclear state is short relative to LC’ the relaxation hamiltonian Je1 (t) has not enough time to produce its full effect during the passage of the nucleus in this state. Accordingly relaxation effects must be reduced by a factor Ln/’Cc. But, when this situation is
realized, we have found that eq. (12) is valid only
under the condition that rot in 1 or rot 1/in.
This mean that in eq. (12) the second term in the deno- minator is then very small compared to the first one,
i. e. relaxation effects will be very small. It is easy to see that the same remarks applies to the time diffe- rential correlations, for which t a few zn T,-
Thus, in any case, the main consequence of the finite lifetime of the nucleus will consist in a reduction of the relaxation effects with respect to what they would
be otherwise, but this reduction will perhaps not be
very easy to demonstrate ; also, as concerns differential P. A. C., one cannot expect to see any oscillating beha-
vior (similar to the one predicted by stochastic theory),
as would be the case if the formulas of (7) were valid
whatever in.
2) APPLICATION TO THE INTERPRETATION OF P. A. C.
EXPERIMENTS. - We shall now sum up the various
inequalities which came into play in the above discus- sions, in order to derive some practical rules for the
use of the different theories in the interpretation of
P. A. C. experiments.
Perturbation theories of relaxation.
Several cases may then be singled out :
b) Observation time shorter than ’te (zn or t - in)
(3) In the case considered here of a purely Zeeman static hamiltonian, the effect of the secular approximation is to decouple the relaxations of the aÀ with different q. An identical
decoupling occurs, even though the secular approximation is -not valid, when the instantaneous fluctuations of Jel(t) are
isotropic [11] or exhibit cylindrical symmetry [5].
Stochastical theories of relaxation.
They are based on simplified physical models but
are valid whatever the correlation time and the obser- vation time.
Choice of the theory to be used in the interpretation
of a P. A. C. experiment.
Let us assume first that 1 which
is the low temperature situation.
Integral correlations.
In these experiments COn ’t’n ’" 1. The observation time is of order ’rn-
- i zn use the G. A. P. theory for isotropic
instantaneous fluctuations, the G. S. H. theory other-
wise
Differential correlations.
Here we may very well have con T. » 1. As for the observation time, it is still of order zn to a few in
- z in and (On « 1, G. A. P. theory (or
G. S. H. theory for anisotropic instantaneous fluctua-
tions),
the theories to be used for both integral or differen-
tial P. A. C. are
(or G. S. H. theory - but anisotropic instantaneous fluctuations are less likely to occur at high T),
- h-2 Jeî > 1 Stochastical theory.
Te > Tn Stochastical theory.
III. Relaxation effects in nuclear orientation expe- riments. - Let us recall that we have demonstrated in (1) that in static N. 0. experiments such effects
can only be observed on levels with « intermediate » lifetimes, i. e. lifetimes comparable with the relaxation
times.
In the following study of relaxation, we shall restrict ourselves to the case where in eq. (1), JC,(t) represents the effect of a fluctuating field with a correlation time much less than Tn, and short enough that one can use perturbation theory. We have already mentioned that
in this case, and within the « high temperature approxi-
mation » (1 h(OnlkB T 1 « 1), the relaxation equations
of the density matrix assume a very simple form :
The demonstration of (13) makes explicit use of the
fact that at « high » temperatures one has :
Our purpose now is to extend this treatment to the very Y low temperature p range, g’ i. e. / hco. k T Î1 > 1. The
notations are the same as in ref. [3] p. 978.
We assume that the fluctuating hamiltonian can be written
Having done all the classical approximations (parti- cularly the secular approximation), and neglecting as
usual the small frequency shift terms, the equation of
motion of the density matrix can be written (in the
interaction representation) :
Let us introduce the Fourier transform of the symmetrical correlation function :
Using the fluctuation-dissipation theorem, it can be shown that :
Finally in the normal representation :
This last equation is valid for any value of the degree 1
of the tensor Ul in eq. (19). It now remains to make explicit the expressions of the Up in the case of a fluc- tuating field (1 = 1) and to relate the corresponding Jl, J- 1, Jo to the correlation times (see ref. [3], eq. (35),
(36), (37)). Then, after completing all the calculations,
we find (4) :
with
It is easily checked that to first order in Iiwn/kB T,
this equation reduces to (13). We notice that in eq. (20)
there appears a coupling between 6k with different k.
Consequently, the relaxation of a given a" k is no more
described by a single exponential ; U" k is a linear combi-
nation of exponentials the time constants of which depend on Tl, T2 and I. As for the coefficients of the linear combination, they are functions of the initial
conditions. The definition of Ti remains the same as at high temperature, but Tl has no longer an imme-
diate physical meaning. In other respects, one should
note that eq. (20) provides some simple relations
between the quantities (u")o at thermal equilibrium ;
these relations could be used to calculate step by step the (0’2)0. (uo)o = /2 I + 1 and (ao), is propor- tional to the Brillouin function.
In standard N. 0. experiments, which are performed
in the absence of radiofrequency, only the uo are
différent from zero. It is then easy to show that (for
(4) Some of these formulas have already been given in ref. [12].
But the number of misprints makes this paper very difficult to use.
q = 0) eq. (21) are strictly equivalent to the classical equations of evolution for the populations pm :
T1 and Tl obey the relations :
As for the ok the solutions of eq. (23) have the general
form :
( j taking 21 values). At T = 0, it is possible to demons-
trate that, setting j = m’ :
As was already mentioned in ref. [1] (sec V, 4c),
with the help of eq. (23) we have been able to give a satisfactory interpretation of some nuclear orientation
experiments performed on Cd109 [13] and on Zn 65 [14]
in Fe. In these cases of diamagnetic impurities in metals, the relaxation of the nuclei very likely proceeds by direct coupling with the conduction electrons, i. e.
with the « reservoir ». Since the reservoir has an
infinite number of degrees of freedom, the notion of a
« correlation time » for JC,(t) loses a part of its inte- rest, and it is better to think in terms of a summation
on a continuous density of final states for the conduc-
tion electrons, in the same way as was done for the pho-
tons in section (II, 2) of (1). One then expects that the
coupling between the conduction electrons and the nucleus will give rise (besides the first order hyperfine
field when the matrix is ferromagnetic) to both a
second order frequency shift and a relaxation. The second order frequency shift may be considered as
the self energy of the Ruderman-Kittel interaction.
As for the relaxation, it can be demonstrated either
directly (eq. (24) and (25)) or with the help of the
fluctuation dissipation theorem that :
in which C is a constant and L- lm X(q".q’, 0153j is the
qq’
imaginary part of the transverse electronic susceptibi- lity at the impurity site [15]. This susceptibility being practically temperature independent, 1/Tl assumes the
form :
which at high temperatures reduces to Ti T = ck
(Korringa law). In the two cases mentioned above,
the application of eq. (23) and (29) has enabled us to determine both ck and co,, from the experimental data.
IV. Radiofrequency effects in angular correlation
experiments. - 1)COUPLING HAMILTONIAN AND ROTAT- ING FRAME. - We assume that the nucleus is submitted to a rotating radiofrequency field :
(HRP)x = Hi cos «Ot) ; ; (HRF)., = Hl sin (cot) (30)
with a circular frequency co in the vicinity of the Zee-
man circular frequency con of the nucleus in state I.
Inside this state, the sum of the Zeeman and R. F.
hamiltonians can be written as :
The resulting equation of motion for the density
matrix is :
If we now set :
(transformation to the rotating frame OXYz), this
becomes :
In this frame, the equation of evolution of the density
matrix is the same as if the nucleus was submitted to the pùrely static hamiltonian :
2) APPLICATION : : RADIOFREQUENCY EFFECTS IN P. A. C. IN THE ABSENCE OF RELAXATION. (’). - Let
us first assume that there is no relaxation. The above
equations show that the density matrix 0 is the solu- tion of :
Jeeff being time independent, this equation is readily integrated, and in terms of the Ok, it gives :
Jeeff has the form :
with
Let d(a) be the rotation around OY which transforms the Oz axis into the direction of n. Starting from
eq. (36), it is easily proved that in terms of the rotation
coefficients d q- (Edmonds (27) p. 55) :
hence : ;
Il
By comparison of (40) and (6), we immediately get the expression of the perturbation factor :
The corresponding quantity for integral correlations is (see eq. (4)) :
(5) This subject has also been investigated independently and in much greater detail by Matthias et al. [16].
