NUCLEAR RELAXATION IN DILUTE MAGNETIC ALLOYS

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NUCLEAR RELAXATION IN DILUTE MAGNETIC ALLOYS

B. Giovannini, P. Pincus, G. Gladstone, A. Heeger

To cite this version:

B. Giovannini, P. Pincus, G. Gladstone, A. Heeger. NUCLEAR RELAXATION IN DI- LUTE MAGNETIC ALLOYS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-163-C1-171.

�10.1051/jphyscol:1971154�. �jpa-00214481�

ALLIAGES DILUES

NUCLEAR RELAXATION IN DILUTE MAGNETIC ALLOYS

B. GIOVANNINI (*), P. PINCUS (**) University of California, Los Angeles, California, 90024

G . GLADSTONE (***) and A. J. HEEGER University of Pennsylvania, Philadelphia, Penna., 19104

RCsume. - On discute et l'on compare les differents mecanismes de la relaxation magnktique des noyaux, par les impuretes magnetiques dans les metaux. On montre que deux des mecanismes proposes preckdemment (1,

peuvent se deduire d'une f a ~ o n unifiee de l'approximation de phase aleatoire ceci conduit a une comprehension claire de la phy- sique mise en jeu et de ses extensions possibles. On montre l'importance des mecanismes qui font intervenir un couplage dipolaire direct et un couplage pseudo-dipolaire (par des electrons de conduction). On compare les donnQs tirees des systemes g M n et G M n . Un strieux desaccord entre la theorie et I'expkrience est mis en evidence.

Abstract.

- The various mechanisms for the magnetic relaxation of nuclei via magnetic impurities in metals are discussed and compared with each other. It is shown that two of the mechanisms proposed earlier [I, 2,3] can be derived in a unified way in the random phase approximation, and this leads to a clear understanding of the physics involved and of the possible extensions. The possible importance of the mechanisms involving direct dipolar coupling and pseudo- dipolar couphng (via the conduction electrons) is pointed out. A comparison is made with the available data on the CuMn and the CdMn systems, and a serious discrepancy between theory and experiment is pointed out.

-

I. Introduction.

In 1963, Benoit, Ide Gennes and Silhouette (BGS) suggested [I] a relaxation mecha- nism for the host nuclear spins in a dilute alloy via magnetic impurities. This relaxation involves a mutual nuclear spin-electron spin flip ; i.e. a real excitation of the impurity. Orbach and Pincus [2] generalized the calculation to include anisotropic systems. In 1969, Giovannini and Heeger (GH) [3] suggested a different relaxation mechanism which involves a virtual excitation of the impurity. Both mechanisms involve the coupling of a given host nuclear spin with a distant impurity via the contact interaction and hence rely on the Ruderman-Kittel-Kasuya- Yosida (RKKY) [4, 51 coupling between the two.

These mechanisms compete with those induced by the dipolar and pseudo-dipolar coupling between a nuclear spin and a n impurity.

The purpose of this paper is to present a unified derivation and discussion of the BGS and G H mecha- nisms, to evaluate the dipolar and pseudo-dipolar mechanisms, and to compare the results with the available experimental results [6, 7, 8, 91.

We shall assume the impurity to act as a well- defined localized spin a t temperatures (and magnetic fields) sufficiently high that Kondo related effects [lo]

may be safely neglected. The treatment here makes no attempt to treat the nuclear relaxation for tempera- tures T < T, even though recent experimental studies demonstrate large effects [l 11. Our goal is the understan- ding of the relaxation in the high temperature (pertur- bative) limit. As we shall see, the situation in this supposedly

simple

regime is far from clear. The work presented here points to some new aspects of the nuclear relaxation measurements in dilute magnetic

(*)

On leave from Institut de

Matihe CondensCe de 1'Uni- versit6 de Genkve, Switzerland.

(**)

Supported

part

National Science Foundation and the

Office of Naval Research.

(***)

Present address,

M. Watson Research Center, Yorkton Heights,

alloys. Of particular interest are questions relating to the dynamic bottleneck effects in the coupled electro- nic system as well as information on the real and imaginary response functions of the conduction elec- tron sea.

The BGS mechanism involves a mutual nuclear- electron spin flip via the RKKY interaction with energy conservation satisfied because of the finite response of the electronic Zeeman transition at the NMR frequency (i.e. the finite ESR linewidth). Thus such a process depends sensitively on the intensity of the transverse imaginary part of the localized elec- tronic susceptibility ~ " ( o ) evaluated a t on, the NMR frequency. Such relaxation studies thus provide a tool for studying ~ " ( o ) far off resonance

in a thermody- namically stable system, this intensity must go to zero at zero frequency ; the model of relaxation towards the instantaneous local field requires that it goes to zero like o l o , (om

electronic resonance frequency).

A second interesting and complicating feature is the question of whether or not << bottleneck

effects should be included in the local moment dynamic susceptibility for this problem. Such effects, as we shall see, can dramatically alter the order of magnitu- de of the nuclear TI 's.

The G H mechanism involves a virtual transition between the spin Zeeman levels of the impurity which is then desexcited by the scattering of a oonduction electron. Such a second order transition has initial and final states which are identical to those for the pure metal Korringa process and thus these two mecha- nisms interfere. It is just this interference term which was calculated in reference 3 in the limit of sharp electronic levels. Indeed we shall show that the G H process is proportional t o the real part of the local moment transverse susceptibility xl(on) in contrast t o xl'(on) for the BGS process. Moreover, the G H mechanism depends sensitively on the spatial varia- tion of the imaginary part of the conduction electron non-local susceptibility, Im Xe'(o, r).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971154

C 1

-

In Part I1 we develop a susceptibility formula which

treats the GH and BGS processes together on the same footing. In Part 111, we analyze the magnetic field and temperature dependence of these mecha- nisms, and compare their magnitude. The dipolar and pseudo-dipolar mechanisms are discussed in Part IV.

A global discussion is given in Part V, together with a comparison with available experimental results.

11. Susceptibility Formalism. - We consider a dilute alloy containing a small concentration of magne- tic impurities, e.g. CuMn and study the longitudinal nuclear relaxation time T I of the nuclear spins of the host (in our example Cu). The Hamiltonian for the system (neglecting dipolar terms for the moment) may be written

x

xn -I- xe + ^Xm + xne + ^{Xme (11.1)}

where the various terms are : a) nuclear Zeeman energy

En

=pionIi'; (11.2)

i

b) impurity Zeeman energy Xm

C A o , s;;

j

c)

electronic energy

Xe

J e e ~

Xint , ^(11.4)

where Xeo is the one electron hamiltonian in the pre- sence of the external field and Xi,, describes the elec- tron-electron and electron-lattice interactions ;

d ) the contact interaction

where A is the hyperfine energy, N the number of sites per unit volume, and o(RJ the conduction electron spin density at site Ri ;

e) the s-d exchange interaction

To begin, we study the relaxation of a nuclear spin at the origin in the presence of a single impurity at the site Rj. The relaxation time TI is given in terms of the non-local dynamic conduction electron suscep- tibility xe(r, r', o) by [12] (we set the volume V

1 every where)

where g, j?, are respectively the g value and magnetic moment of conduction electrons. In the absence of impurities and for a free electron gas

N(0) being the density of states at the Fermi surface per unit volume. We consider the system without impurities to be translational invariant, i.e., xe ^{is a}

function of r - r', whereas with impurities xe depends separately on r and r'. Applying expression (11.8) to equation (11.7) gives the standard Korringa result.

Moriya [12] has shown how this result is enhanced by electron-electron exchange.

The problem at hand is to compute xe(r, r', on) in the presence of magnetic impurities. This problem should be solved by some well defined and consistent many-body analysis similar to the one used in the EPR problem [13]. Here, however, we shall employ a n?olecular field approach in terms of the conduction electron susceptibility in the absence of magnetic impurities xe(r, o ) and the full impurity transverse susceptibility ~ ( w ) . ze(r, r' o ) is given by the response of the electron gas to a transverse field h-(r, w)

< e-(r, I ) >

^(gj?)-' I ^{f (r,}

^{t - t')}

x h-(r', t') dt' dr' (11.9) taking

-

h-(r, t) = h6(r) eUiw' (11.10)

The local conduction electron spin density is then given in the molecular field approximation by

< 07(r, o ) >

xe(r, o ) h6(o - ^Z)

where the second term represents the effective conduc- tion electron magnetization arising from the trans- verse part of the impurity magnetization. Similarily,

where ~ ( o ) is the transverse magnetic response of the local spin. In the above expressions we have assumed that the localized and conduction electrons have iden- tical g values (for simplicity). Combining (11.1 I), (II.12), and (II.13), we obtain for the conduction electron response,

xe(r, 0, o )

This equation is easily solved to give xe(O, 0 , ~ )

(11.15) describes an impurity enhancement of the conduction electron susceptibility. Such an enhance- ment has already been discussed in other terms by Kim [14]. It is straight-forward to generalize this calculation for a finite concentration of impurities (see Appendix A).

The vanishing of the denominator (see equation

A. 2) indicates the possibility of magnetic ordering

NUCLEAR RELAXATION IN DILUTE MAGNETIC ALLOYS C 1

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of the system. In this work we shall assume concen-

trations sufficiently small that the system is far from any such instability and we shall therefore retain only the terms linear in concentration. This is clearly the physical situation in dilute alloys with impurity concentrations less than a few hundred ppm.

Defining the functions

= (gSN(0))-'(4 n ~ , ) - ' Re xe(r, w,) ]

and

and noting that

Im xe(r, on) < Re xe(r, o n )

the imaginary part of the conduction electron suscep- tibility now becomes (see equation A.4)

Note that for free electrons (r

1 r I)

(2 kf r) cos (2 k, r) - sin (2 k, r) ao(r)

(2 kf

(I1 .19) sin2 (kf r)

DO(d =

(kf rI2

The functions F0 and Go corresponding to this case are given in Appendix B. Equation (11.18) is our central result and gives T1 when combined with (11.7). The various contributions to T I appearing in (11.18) may be identified in the following manner. As has already been pointed out, the first term is nothing but the Korringa process (including many body corrections) for the host in the absence of impurities ; the latter terms proportional to the impurity concentration will be shown in the next section to be the GH mechanism (that part proportional to Re ~ ( o , ) and the BGS mechanism (that part proportional to Im ~(o,)).

111. Discussion of the BGS and GH mechanisms.

- In order to discuss (11.18) we use for the localized moment susceptibility ~ ( o ) the from describing relaxa- tion to the instantaneous field [13]

where z2 is the appropriate local moment relaxation time. Although in the simple model of relaxation towards the instantaneous field

is a constant, we will allow for a possible dependence on o.

Since o, << a,,, we obtain from (11.7) and (11.18)

It is important to note that in obtaining (111.3) we have explicitly used relaxation toward the instan- taneous field. This results in a drastic reduction of the low frequency response, Im ~(w,) by a factor of (w,/om) as compared with that obtained from a sym- metric Lorentzian absorption. The final result, howe- ver, is in agreement with that obtained from the simple Golden Rule calculation of BGS. The results of the present calculation thus emphasize once again [15] the importance of relaxation toward the instan- taneous field especially with regard to the response way off resonance.

Taking the ratio of (111.2) to (III.3), we find (Ti- = _1 FG-0) . ^hz;' _{(1 + (w,} T ~ ) ~ ) . ^(111.4)

s 4 G(r0) Ef

This ratio is clearly a sensitive function of the value of the localized moment reIaxation time z,, as well as the spatial functions F and G. We note that in equa- tions (111.2) and (111.3) the relaxation time zz enters explicitly only into the BGS term and there through the imaginary part of the local spin response, Im ~(w,).

The first question which arises is whether

should be the bottlenecked relaxation time as measured in the ESR experiments or the much shorter non-bottle- necked time [16]. Although one might intuitively expect the bottlenecked value to be appropriate since the BGS mechanism involves an absorption of energy by the local spin system, the question is not at all clear. The essential point is that the bottleneck is a property of the uniform combined resonance of the conduction electrons plus localized impurities. On the other hand, the case of interest here is equivalent to a resonance induced by a magnetic field acting solely at one point of the system (see equation 11.10) and acting only on the conduction electrons. An alter- native way of expressing this is to note that although the total magnetization commutes with the isotropic s-d exchange (thus yielding the bottleneck pheno- menon) local fluctuations may be very large. These arguments suggest that the local response might be given by the non-bottlenecked impurity susceptibility.

An analogous problem formulated within the Hase- gawa model is discussed in Appendix C to illustrate this point.

An added complication is that, as remarked above,

2,

can be an explicit function of the frequency. There

are two ways that one can imagine an expiicit fre-

quency dependence of the local spin relaxation time ;

first a frequency dependence involved may come from

the Korringa process itself, and second, a frequency

dependence of the bottleneck factor, if the bottleneck

C 1 - 1 6 6 B. GIOVANNINI, P. PINCUS, G. GLADSTONE AND A. J. HEEGER

plays a role. Orbach and Spencer [13] have shown,

however, that the basic Korringa mechanism for local spin relaxation is essentially frequency independent.

The question of a frequency dependence for the bottIe- neck effect is somewhat less clear. In fact, if the conduc- tion electron spinlattice relaxation mechanism involves inelastic scattering the bottleneck may be more effec- tive a t low frequencies ; i.e. z2(wn) > z , , , , where r

, , ,

, is that measured directly in the EPR experi- ments.

The question of the bottleneck and of its possible frequency dependence therefore can only be answered by a detailed microscopic analysis, which we hope to carry out in the future. Our goal here is to attempt to clarify these questions by comparison of the calcu- lated contributions with experiment (see Section V).

Finally, since again we are looking a t a very non- uniform problem, z , can be influenced by the impu- rity-impurity interactions. This has been shown explicitly to occur in ordered metallic systems by Silbernagel et al. [17]. The order of magnitude of

J Z

this effect is

- ^{C -}

(which is of h E.

the order of the observed .r;) and is essentially tem- perature independent.

Before going into a direct comparison with the experimental data, we discuss the behavior of the two processes as a function of the parameters T and H.

i) THE GH

- From the result, equa- tion (111.2) one sees that

( T ~ - ' ) ~ ~ - ^{< S' > .}

_H T ^(111.5) is a function of HIT which for

4 k, T is indepen- dent of H or T and for Aw, >> ^k, T varies as U H .

ii) THE BGS

- AS can be seen from (111.3)

At fields such that w, z 9 1

(TI-')BGs - ^{< Sz > 7} _H T ^.

^{' ( H , T) . (111.7)} At low fields ( a , z) 4 1

(TI-'),,, < ^SZ > - T r2(H, T) (111.8) H

which is the same dependence as the GH term except for the explicit dependence of z2 on H and T.

IV. Dipolar and Pseudo-Dipolar Contributions to the Relaxation Rate. - The hamiltonian describing the coupled localized electron-conduction electrons- host nuclear spin systems as given by equations 11.1-6 neglects the magnetic coupling between the dipole moments of the three spin species. The total hamiltonian describing the coupled system is

3e.r=x+x,D,+x,D,+x:m (IV.1) where R is given in equation (JI .I). As before e

con- duction electron, m = magnetic impurities, n = host nuclei.

The dipolar coupling between two magnetic mo- ments g , BS, and gz /3SZ separated by r . c ( r is the dis- tance between the two and /; is the unit vector in the appropriate direction) is given by

(IV .2) Let us first discuss the direct dipolar term x:,

whose importance in relaxation is well-known, and let us consider first, as in Part 11, the influence of an impurity at Rj on the relaxation of a nucleus at the origin.

The largest contribution to the host nuclei relaxa- tion rate comes from the anisotropic terms of the dipolar coupling in which the z-component of the impurity spin is conserved and is easily calculated to be

where r,(wn) is the longitudinal relaxation time of the local spin measured at the NMR frequency.

At low fields the direct impurity dipolar relaxation rate is considerably smaller than the BGS rate (equa- tion 111.8). Taking the free electron value for u(Rj) (11.19) and the hyperfine coupling constant appro- priate for Cu one has in the limit where H / T is small and w,z2 4 1

I

s

JN(O) '

(TI-

(on) ( 7 )

With increasing magnetic field ( T ; ' ) , ~ ~ falls as H-2 while (Ti-'), is magnetic field independent as long as the field is sufficiently small that saturation of the impurity magnetization is not important. At higher fields, i.e., X = gPHlkT > 1,

so that the dipolar rate goes exponentially to zero.

This behavior is to be contrasted with that of the GH mechanism (equation 111.2) which gives a rate decreasing only as H-' in the high field limit. Finally in comparing the dipolar and GH mechanisms, we note the

enters explicitly into the dipolar rate expression. This implies, as discussed above that the magnitude of the dipolar rate depend on whether the bottlenecked or non-bottlenecked time is used.

In addition to the direct dipole-dipole mechanism,

there are three conceivable indirect mechanisms

which we denote symbolically by X,,.XYm, X;~.X,,

and ~ e f l , . ~ : ~ (see equations 11.5 and 11.6). The

first, X,,,.X;~ should be discussed within the for-

mula (11.7). The difficulties encountered when one

attempts to calculate the dipolar relaxation rate of

the conduction electron are quite severe. Such a

calculation depends sensitively on the shape of the

Fermi surface and the relative amounts of s-p and d-

NUCLEAR RELAXATION IN DILUTE MAGNETIC ALLOYS C 1

-

wave character of the conduction electrons at the Fermi surface, as well as on the conduction electron spin susceptibility. It is easy however to see that this first process is small compared to the one denoted by

xYe.&,. If the conduction electrons wave function within the impurity do not differ greatly from their wave function at an observable host nucleus, one readily sees that the first process (X,, x: ,) is smaller

J

Pn

than the second ( X f ; k . ~ , ~ ) by a factor

- -- lo-', A u,

and we shall therefore neglect it. The third 'process is smaller than the second by a factor

We are left with the term x?,. x,,, ((( pseudo-dipolar

coupling). The difficulties encountered in the estima- tion of this mechanism are essentially the difficulties involved in the calculation of the dipolar relaxation rate in the pure host. The mechanism we consider is then the enhancement of the dipolar relaxation rate due to magnetic impurities, and is completely analo- gous to the BGS and GH enhancements of the contact relaxation rate.

In both cases the magnetic fluctuations in the conduction electron sea are enhanced by the s-d exchange interaction. The host nuclei are relaxed via the Fermi contact interaction in the BGS and GH mechanisms while for the

pseudo-dipolar

mecha- nism relaxation is accomplished via the conduction electron-nuclear spin dipole interaction.

Bloombergen and Rowland [IS] have considered the relative importance of two quite analogous pro- cesses in producing homogeneous line broadening in a pure host. The analog of the BGS mechanism is the RKKY coupling between two host nuclei. This cou- pling is mediated by the Fermi contact interaction between nuclei and conduction electrons at each of the host nuclei. The analog of the pseudo-dipolar interaction is one which utilizes the contact coupling between the conduction electrons and one of the host nuclei while the second nucleus of the interacting pair senses the conduction electrons via the dipole-dipole coupling. It is quite straightforward to extend their calculation to include the pseudo-dipolar coupling between nuclei and magnetic impurities and compare the amplitude of this coupling with the Yosida cou- pling between them (which leads to the BGS enhance- ment of TT1). One finds a pseudo-dipolar interaction of the form

PD ^{A A}

Xmn=CBijIi.(Sj-rij(rij.Sj)) (IV.5)

i.j

where the quantities Bij consist of momentum integral over products of trigonometric functions and conduc- tion electron Bloch functions. When the impurity and the nucleus under consideration are separated by several lattice constants, Bloombergen and Rowland showed that the Bij 's look very much like the product of the contact potentials and the RKKY spacial func- tion

Bij -- < B > cos (2 kf rij + qB)/(2 kf rij)3 . ^(IV.6)

This shows that the pseudo-dipolar couplipg contri- bution to TI-' has the same temperature and field

dependence as the dipolar contribution (equations IV. 3 and IV.9) and will only change its magnitude.

For TIzo5 Bloombergen and Rowland indicate that the pseudo-dipolar potentiel < ^B > is perhaps 30 0/, of A the contact potential. If the pseudo-dipolar poten- tials < ^B > remain comparable to the contact poten- tials A for the Cu host then it's clear that the pseudo- dipolar relaxation rate would be several orders of magnitude larger than the simple dipolar relaxation rate. As Cu has an isotopic mass which is one-third that of TIzo5, this is unlikely.

In absence of detailed information about the cha- racter of the electron wave function at the Fermi sur- face, the best we can do is to view the relative strength of the pseudo-dipolar relaxation mechanism as an experimental parameter

if the high field relaxation enhancement is unequivocally related to dipolar- type terms, one can easily determine the magnitude of the pseudo-dipolar relaxation by simple substrac- tion.

Finally we note that the relative importance of the BGS and pseudo-dipolar mechanism in enhancing TI-' at low fields [om z2(on) 4 11 is determined by the relative sizes of the contact and dipolar relaxation rates in the pure host. One could perhaps determine this latter ratio from relaxation rate and Knight shift studies in the pure host, as the dipole coupling bet- ween conduction electron and the nuclei does not contribute to the observed Knight shift.

To calculate the relaxation time due to a finite con- centration of impurities, we proceed as in section 11, (See Appendix A) and obtain (o,

<< 1)

Spin diffusion plays little or no role in determining the macroscopic relaxation rate. This is essentially because magnetic dipole and electric quadrupole broadening and wipe-out (for host nuclei with I 2 1) prohibit a significant number of nuclei close to magnetic impuri- ties from participating in macroscopic relaxation. As a result the inner cutoff of the lattice sum due to spin diffusion is perhaps one-tenth the size of the magnetic dipole and electric quadrupole inner cutoffs.

Defining the wipe-out number no as the number of nuclei within the cutoff radius r,

equation (IV .7) becomes

At low fields no is dominated by electric quadru- pole wipe-out. From Lumpkin's NMR intensity studies, it can be shown that the wipe-out numbers appropriate to the low field T I measurements are

no = 525 ... T

1.3 OK

= 470 ... ^T

^2.2OK.

C 1

-

At high fields the situation is complicated by the

~nhomogeneous line broadening of the host NMR line which indicates that nuclei near (but not too close to) an impurity continue to participate in macroscopic relaxation even though the impurity magnetization and thus the

local >> Knight shift are increasing with decreasing temperature (at constant H). We shall assume a wipe-out number corresponding to a cutoff ro equal to 3.5 times the nearest neighbor distance

no = 300.

This number is found to be appropriate for CuFe above T, as well as about right for G M n at lowxm- perature. We can now investigate the agreement bet- ween theory and experiment.

V. Comparison between Theory and Experiment.

- For convenience we first rewrite the BGS, GH, and dipolar formulas, with one additional approximation

We replace the trigonometric function occurring in a0 and p0 by their average value ; i.e. we set

Although the effect of magnetic impurities on the host nuclear relaxation can be observed in the early work of Sugawaralg, quantitative measurements at temperature sufficiently greater than the Kondo tem- perature are available solely on the system CuMn and the system CdMn (the Mn spin is in both cases close to 512). ~ e t s first look at the G M n data.

Lumpltin [6] measured the effect of 10 ppm Mn impurities on the C d 3 TI at very low fields, i.e.

H < 200 Oe. The high field regime was studied by

Levine [7] for a sample containing 110 ppm. Thus although the experimental picture is certainly incom- plete, the two important regimes have been studied.

The high field data of Levine show an impurity contribution to the relaxation rate, the magnitude of which is independent of H and T for magnetic fields as high as 12 kOe and a t temperatures as low as 1.4OK (gpH/kT -- 1). This is in agreement with the predictions of the GH mechanism which is explicitly independent of H and T a t low fields and only weakly dependent on H as the magnetization begins to satu- rate. A much stronger field dependence would be expected for the dipolar contribution even assuming

z,(o,) to be independent of H and T. The quantity of interest for the field dependence is

< S: > - < ^S,>2

e-X/21-2 -

where S is the impurity spin and X

gp,H/kT.

Evaluation for the spin 512 case appropriate to Mn predicts a decrease in the dipolar contribution by a factor of four on going from 4,2OK and 7 kOe to 1.4 OK and 12 kOe. The high field data for CuMn are in direct contradiction to this as no suchfield and temperature dependence has been observed. Since the data show no HIT dependence, we must conclude that the dipolar contribution is not dominant. Although the lack of field and temperature dependence in the CuMn data is quite convincing, extension of the expe- - riments to larger values of HIT are surely needed in order to the absolutely certain.

Let us pursue this argument somewhat further. The magnitude of the dipolar contribution can be estima- ted from equation (V.3) using the values

appropriate to Cu. One finds

(TI-'),-- 1.8 x 1013cz1(o,)(s-1). (V.5) The above expression is the (HIT) independent value valid in the high temperature limit (gpH/kT < 1).

This is to be compared with the experimental result (T; I),,,

4,5 x 103 (S

( v - 6 ) As the dipolar term predicts too strong a field depen- dence and therefore seems not to be dominant, we conclude

zl(oN) 4 3 x 10-lo

(V - 7) The bottlenecked ESR linewidth gives a value of z, - ^lo-'. The comparison thus seems to imply that z,(w,), which appears in the longitudinal corre- lation function, is not bottlenecked.

On the other hand Lumpkin's low field data are well described by the expression

with

the impurity concentration and

-- s (temperature independent in the limited range studied).

In this case the implied electronic correlation time is in agreement with that found in the ESR measu- rements. Taking Altz = 1.3 x 10lOs-I from Knight shift and susceptibility measurements, using the free electron values for [N(0)Ef/N]

312 ; and N - l k:

3 n2, and taking (JN(O)/N)

0.15 appropriate to CuMn, we obtain

-

which must be comppred with the experimental result

given above. Although the theoretical prediction is

somewhat large in magnitude (roughly an order of

magnitude) the agreement is not bad considering the

NUCLEAR RELAXATION I N DILUTE MAGNETIC ALLOYS C 1

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approximations made. Certainly the field and tempe- rature dependences are correct.

These data therefore suggest that the BGS mecha- nism is dominant at low fields, and that the bottle- necked relaxation time should be used in the expres- sion (V. 1). The experimental suggestion that the bottle- necked z2 enters into the BGS mechanism whereas a much shorter

suppresses the dipolar mechanism is clearly of interest. Such a duality is perhaps not unreasonable as the BGS mechanism involves a true absorption of a quantum of energy in the tail of the combined electronic response function whereas the dipolar term arises solely from longitudinal fluctua- tions of the impurity spin. Thus the former could be bottlenecked and the latter not. However, this inte- resting explanation can only be understood in detail from a thorough microscopic treatment of the problem.

Having argued that the dipolar mechanism is to be ruled out at high and low fields we conclude that :

(i) The BGS mechanism dominates for corn z, 4 1.

(ii) The GH mechanism dominates for w,

9 1.

We demonstrated above that the magnetic field and temperature dependences are consistent with this interpretation, and that the BGS mechanism is of the proper magnitude. However there is a funda- mental discrepancy between theory and experiment.

Identifying Levine's high field data with the GH mechanism and Lumpkin's low field data with the BGS mechanism leads to the experimental ratio :

On the other hand the theoretical ratio is given expli- citly in (111.4)

{s lTheor.

^{( v . 11)}

The disagreement by a factor of lo3 is all too evident.

I t seems that there are two choices with regard to the large HIT region :

a) The dominant mechanism is dipolar. The magni- tude is approximately correct provided the bottle- necked z is used. However the HIT dependence seems experimentally much too weak.

b) The dominant mechanism is GH. The dependence upon HIT is correct, however the magnitude is not in agreement with simple theory based on a free electron model with a delta function interaction.

The HIT dependence is the crucial point. The expe- rimental data on CuMn presently strongly favor (b) above. However another factor of two in the HIT range would be definitive.

NMR relaxation studies have been recently carried out in the CdMn system [8, 91. The low field data are consistent with the BGS mechanism both in functional dependence on HIT and in order of magnitude. The high field data again show and (HIT) dependence con- sistent with the GH mechanism. However, these authors argue that their accuracy is insufficient to rule out the dipolar contribution, and they favor this dipolar explanation for the obvious reason that it gives the correct magnitude.

Finally, we note the anomalous low temperature T , data in CuCr reported by Gladstone [l I]. Since the dipolar mechanism should not be sensitive to low temperature correlations in the electron sea due to the Kondo effect, these CuCr data also argue against the dipolar contribution:

The largest uncertainty and most likely source of error in the theory would seem to be in the range functions a(r) and P(r) for the real and imaginary parts of the conduction electron susceptibility. Whereas the effects of impurities [20, 211, band structure [22]

and exchange enhancement [21] on a(r) have been explicitly investigated previously, both theoretically and experimentally less is known about the effect on P(r). Furthermore the theoretical estimates would be significantly altered by including the proper q-depen- dence of the exchange J(q). The resulting form of the spin density oscillations are indeed quite sensitive to the form of J(q), particularly in the vicinity of the inner cut-off, and a large error could be introduced in this manner. Nevertheless, in spite of such possibi- lities, it is remarkable that the errors in a(r) and P(r) are so large that the factor of lo3 discrepancy dis- cussed above results. As these range functions are fundamental to our understanding of the magnetic response in metals, this question and the associated discrepancy is of considerable importance.

In the future we hope that a systematic perturbation theory for this problem will be carried out as a func- tion of the direct spin-lattice relaxation rate in order to shed some light -on the very interesting bottleneck questions. The theory should subsequently be extended into the Kondo regime T < Tk, where recent experiments [ l l ] have shown a large enhan- cement of TT1 as compared with extrapolations from the regime T % Tk. However, before one can hope to do this, we must obtain an understanding of the processes in the

simple >> region above T,.

Appendix A

When many impurities are present, equation 11.14 becomes

which cannot be solved in closed form without per- forming an average over the impurity sites. Perfor- ming this average for the lowest class of averaging process [24] (averaging independently over each impurity) gives

(the volume of the specimen is set to unity). Higher

classes of averaging processes can be shown to give

C 1

-

a small contribution. Equation A.2 diverges when

the system becomes ferromagnetic, and then the nuclear relaxation rate becomes also singular. The equation (13) of reference 3 (<( avec un peu d'audace

turns out therefore to be essentially correct.

In the low concentration limit, far from a magnetic transition, we can expand A.2, and get

x

d3r(Xe(r, o))? ( A . 3) This last formula can be made more accurate by consi- dering that only impurities from a lower cut-off r,, which is at least the interatomic distance, contribute.

The value to be given to ro is discussed in section IV.

We therefore rewrite A . 3 as

x " d3r(xe(r, o ) ) . ( A . 4)

Po

The fact that both the G H ahd BGS effects are linear in concentration (in contradiction to what was sta- ted in reference 1) can be made obvious by first consi- dering the average over all nuclei with one impurity present, and this reproduces equation A . 4 in a direct manner.

Appendix B

Although the free electron limit has a rather unrea- listic meaning, we state for completeness the values of GO(ro) and F0(r0), as defined in equations 11.17 and 11.19, one has

where

" (X cos X - sin x)'

-

dX

xo x6

" (X cos X - sin X) sin2 X/2

dX (B. 4)

xo

x4

one finds

1 cos 2 x 0 cos Xo 1 1 - -

6 ( 4 ~ ; x", ) ^{8 ~ ;}

Where

"

X c ( x 0 )

-y- d x

are tabulated functions.

Appendix C

In order to illustrate the fact that the bottleneck question in our problem is not straight-forward, we discuss a somewhat simpler problem in the Hasegawa scheme. Suppose we have two spins systems whose magnetizations are Me and M,, and suppose we want to calculate X: in analogy to equation 11.14. ( ~ b t e that this corresponds to X: (q

O).) We then define X:

~ , f / h + where hC is a field acting only on the system e. Neglecting relaxation towards instantaneous field [15] in the Hasegawa equations, and taking the complete bottleneck limit (Tel = co, where T,, is the conduction electron lattice relaxation time) we have the system of equations

(8, + ^{CO) M,+}

cm ^M:

⁰

(ce + ^{W) M:} - Ce M,f

h+

with

and the corresponding quantities for qe, ce ^and ce

are obtained by permuting e and m everywhere.

One then obtains

+ M: - nelge + o

Xe = + -

_{h 1 -} . ^{(C. 3)}

Ce

CmI(&e + o ) (&m + a )

Equation C.3 must now be compared with equation

A. 2. This leads to the identification

NUCLEAR RELAXATION IN DILUTE MAGNETIC ALLOYS C 1

-

In the denominator, one then sees that cN,(R) should Since we are in the complete bottleneck regime, a be identified with the expression simple-minded bottleneck interpretation of ~ [ w ) would

? - 2 f, I,

lead to

because I = J/Ng2 p2. ^It is obvious from (C.4) that this is not the case.

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