ARTICLE IN PRESS

Physica B 404 (2009) 914–916

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/physb

The spin lattice relaxation of 8Li in simple metals

M.D. Hossain a, H. Saadaoui a, T.J. Parolin d, Q. Song a, D. Wang a, M. Smadella a, K.H. Chow b,M. Egilmez b, I. Fan b, R.F. Kiefl a,e, S.R. Kreitzman c, C.D.P. Levy c, G.D. Morris c, M.R. Pearson c,Z. Salman c,f, W.A. MacFarlane d,�

a Department of Physics, University of British Columbia, Vancouver, BC, Canada V6T 1Z1b Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2G7c TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, Canada V6T 2A3d Chemistry Department, University of British Columbia, Vancouver, BC, Canada V6T 1Z1e Canadian Institute of Advanced Research, Canadaf Clarendon Laboratory, Department of Physics, Oxford University, Parks Road, Oxford OX1 3PU, UK

a r t i c l e i n f o

Keywords:

Korringa law

Spin relaxation in metals

26/$ - see front matter & 2008 Elsevier B.V. A

016/j.physb.2008.11.157

esponding author. Tel.: +1604 822 6866.

ail address: wam@chem.ubc.ca (W.A. MacFarl

a b s t r a c t

We report the modification to the linear temperature dependence of the Korringa nuclear spin–lattice

relaxation rate of an implanted NMR probe in silver, as it makes a thermally activated site change. We

develop a simple model of this phenomenon, which is found in a number of metals including Au and Nb.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

The nuclear spin–lattice relaxation rate l ¼ 1=T1 in nonmag-netic metals is often due primarily to spin flip scattering from theconduction electrons, the so-called Korringa relaxation [1], withits characteristic linear temperature dependence

l ¼pA2

hf

_r2ðEF ÞkT ,

where, in the simplest model, Ahf is the contact hyperfine couplingof the conduction electrons (having density of states at the FermiEnergy rðEF Þ) to the nucleus. For light implanted ‘‘nuclei’’ like themuon and 8Li the coupling is generally small and l is conse-quently almost always too slow to measure with mSR; however,this is not the case with the much longer lived b-NMR probes(8Li has a mean lifetime t ¼ 1:21 s), where Korringa l has beenmeasured in a number of metals, including some where theconventional host lattice NMR is difficult or impossible due to thelack of a sensitive (high gyromagnetic ratio) nucleus, e.g. Pd andAu [2,3]. Korringa relaxation is an important means of character-izing a metal and deviations from it have been very useful in thestudy of metals. For example, the Hebel–Slichter coherence peakin l that occurs at the superconducting transition can be used todistinguish a conventional s-wave superconductor from moreexotic pairings [5].

Another common occurrence for 8Liþ implanted in metals is atemperature dependent site change. At low temperature, the

ll rights reserved.

ane).

8Liþ is trapped at a metastable interstitial site, but at highertemperatures, it makes a thermally activated transition, into anearby vacancy and becomes substitutional [2,3,9,11]. In factinterstitial Li diffuses appreciably in Ag only at much highertemperatures than the observed transition (450 K [6]), and thetransition may instead be due to the onset of vacancy mobility inAg (Stage III annealing [7]). The main objective of this work is toelucidate the modification to the simple Korringa temperaturedependence caused by such a site change.

2. Experimental

The data described here were taken on two samples of Ag, a500 nm film evaporated on an SrTiO3 substrate and a 25mm foil.The b-NMR experiment was carried out at the ISAC facility atTRIUMF using a beam of highly polarized radioactive 8Liþ with atypical rate R0 � 107=s into a beam spot about 3 mm in diameter.The 8Li nucleus has a spin I ¼ 2, g ¼ 630:15 Hz=G, and an electricquadrupole moment Q ¼ þ31:4 mb. Details on the polarizer andspectrometer can be found in Refs. [8,9]. Resonance measure-ments were carried out with a continuous beam and a continuouswave radio-frequency (RF) magnetic field applied transverse tothe spin polarization. Spin–lattice relaxation measurements werecarried out with a pulsed beam of length D with the RF field off.Data were collected over a range of applied field H0. Examples ofthe spin–lattice relaxation data in the Ag foil are shown in Fig. 1,and their analysis is discussed in the next section. An exampleof resonance data is shown in Fig. 2, where we show theresonances with the beam polarized in both directions (parallel

ARTICLE IN PRESS

0Time(s)

0

0.1

0.2

0.3

Asy

mm

etry

4 K100 K150 K250 K

4 8 12

Fig. 1. The temperature dependence of the 8Li spin lattice relaxation in a silver foil

at H0 ¼ 150 G. The beam pulse length is D ¼ 4 s.

18910-0.5

-0.4

-0.3

Asy

mm

etry

Frequency (kHz)

2b

Ag film, 3 T, 160 K

18915 18920

Fig. 2. The definition of the off resonance baseline asymmetry b in a continuous

wave resonance measurement. The asymmetry is integrated for 1 second with the

RF field applied at the frequency given by the ordinate. The data show the two

resonances from the two sites in Ag. The two curves correspond to the two

helicities of the polarizing light.

0.4

0.3

0.2

0.1

0.0

λ (S

-1)

0 200 300100Temperature (K)

Foil 150 G50 nm film, 3 T, Morris et al.50 nm film, 3 T, Scaled Baseline

Fig. 3. The temperature dependence of the 8Li spin–lattice relaxation rate in two

samples of Ag. The open circles are from values of the average off resonance

baseline asymmetry scaled as described in the text. The fit curve is the activated

site change model described in the text.

M.D. Hossain et al. / Physica B 404 (2009) 914–916 915

and anti-parallel to H0) in order to clearly define the off-resonancebaseline asymmetry b (see also Ref. [9]).

3. Theory and discussion

When a continuous beam of spin-polarized probes of lifetimet is implanted at a constant rate, and if the implanted probes relaxas pðtÞ ¼ expð�ltÞ, a dynamic equilibrium polarization is estab-lished at a value

p ¼1

t

Z 10

pðtÞ e�t=t dt ¼1

ltþ 1.

In a b-NMR resonance measurement, using a continuous beam,when the RF magnetic field is off resonance (and has therefore noeffect on the polarization), we obtain the experimental ‘‘baseline’’asymmetry, b ¼ ap (see Fig. 2), where a is the experimental factorthat accounts for the asymmetry of the beta decay, the geometryof the detectors, etc. Therefore, the baseline is determined byl, with l ¼ 0 yielding the maximal bð¼ aÞ. One can easily invertthe above to find

l ¼a� b

tb, (1)

which has the single unknown parameter a.The spin lattice relaxation rate in the silver film was measured

in H0 ¼ 3 T in time differential mode using short (D ¼ 0:5 s) beampulses [9]. The resulting spectra were fit to a single exponential toyield the Korringa rate (stars in Fig. 3). The initial asymmetryusing such short pulses is high, but the statistics is rather low, as alimited number of probes are implanted. More recently, wemeasured l in a 25mm Ag foil using much longer beam pulses(Fig. 1), and fit the time dependent asymmetry during and afterthe pulse to yield a more complete data set through the sitechange at about 170 K (filled squares in Fig. 3). The slopes of thesetwo measurements are similar, but slightly smaller for the foil.The foil data, however, clearly reveal the effect of the site changebetween about 110 and 170 K, l executes a crossover from a steeplow temperature line to a shallower high T slope. We also includel values from the resonance baseline of Morris et al. [9] usingEq. (1) at 3 T (open circles), with the constant a ¼ 0:195 chosen tomatch the foil at 100 K.

We now develop a simple theoretical model of this crossover.Let us define the beam polarization from the optical polarizeras p0. In terms of the populations pm of the 2I þ 1 magnetic sub-levels of the probe nucleus p0 ¼ ðSpmÞ=I. We account for twoinequivalent sites (1 and 2), each with characteristic singleexponential relaxation rate l1; l2, and include a thermallyactivated transition 1! 2. We assume implantation results ininitial probe fractions at each site f 1 and f 2 ¼ 1� f 1 that areindependent of temperature, and that the rate of site change isn ¼ n0 expð�Eb=TÞ, where n0 is the attempt frequency for theenergy barrier of height Eb (in K). We consider the polarization of

ARTICLE IN PRESS

M.D. Hossain et al. / Physica B 404 (2009) 914–916916

a probe nucleus implanted at time s0 and decaying later at time s,

pðs; s0Þ ¼ pðt ¼ s� s0Þ ¼ p1ðtÞ þ p2ðtÞ. (2)

The polarization for the two sites follow the rate equations:

_p1 ¼ �l1p1 � np1, (3)

_p2 ¼ �l2p2 þ np2, (4)

subject to the initial conditions p1ð0Þ ¼ f 1p0 and p2ð0Þ ¼ f 2p0,which has solution:

pðtÞ ¼ p1ðtÞ þ p2ðtÞ

¼ ½Me�ðl1þnÞt þ ð1�MÞe�l2t�, (5)

where

M ¼p0f 1ðl2 � l1Þ

ðl2 � l1Þ � n. (6)

Experimentally, to measure 1=T1, we observe the time-resolvedexperimental asymmetry AðtÞ, during and after a pulse of beam,represented by Rðt0Þ ¼ R0ðyðt0Þ � yðt0 �DÞÞ, where y is the unit stepfunction, and R0 is the constant instantaneous rate. For a genericpolarization function pðtÞ, the observed asymmetry (averaged overthe incident pulse and probe lifetime) is

AðtÞ ¼A

NðtÞ

Z t

�1

Rðt0Þ e�t0=tpðt0Þdt0, (7)

i.e. the Laplace transform of the product of the square beam pulsewith the polarization function. The number of probes in thesample at time t is

NðtÞ ¼

Z t

�1

Rðt0Þ e�t0=t dt0, (8)

which normalizes the previous expression, and A is the propor-tionality constant between the observed asymmetry in thedetector count rates and the instantaneous average polarizationof all the probes in the sample. Substituting the biexponential twosite relaxation function from Eq. (5) into Eq. (7), we find thatduring the pulse (0ptpD)

AdðtÞ ¼M

l01t1� e�l

0

1t

1� e�t=t

!þ

1�M

l02t1� e�l

0

2t

1� e�t=t

!, (9)

where l01 ¼ l1 þ nþ t�1 and l02 ¼ l2 þ t�1. After the pulse (t4D)the asymmetry follows a simple biexponential:

AaðtÞ ¼M

l01t1� e�l

0

1D

1� e�D=t

!e�ðl1þnÞðt�DÞ þ

1�M

l02t1� e�l

0

2D

1� e�D=t

!e�l2ðt�DÞ.

(10)

Notice that the amplitudes of the two components after the pulsedepend upon the relaxation rates and D such that if onecomponent has a relaxation rate much faster than D�1, then itscontribution to the signal after the pulse is highly suppressed.

Though the solution above is biexponential, the data appearsingle exponential after the pulse, probably because the tworelaxation rates are similar. Thus we assume the measured singleexponential relaxation rate represents the weighted average of thetwo components with weighting factors given by the relativeamplitudes in Eq. (10). The resulting expression is algebraicallycumbersome, but not difficult to evaluate, so it was used in a leastsquares fit to the data (black squares only) in Fig. 3. The data fit

reasonably well, except near the site transition, where the modelis much sharper in temperature than the data. It is possible toobtain a better fit by allowing the attempt frequency n0 to be verylow (on the order of kHz), which is unphysical for an ion in mostcrystal potentials. More likely, the site change is characterized bya distribution of energy barriers and attempt frequenciescorresponding to a variety of configurations for the interstitial Liand nearby vacancies. This would yield a smeared-out transitionas observed.

However, the fit shown corresponds to n ¼ ½1012 s�1� expð�½4040 K�=TÞ, with Korringa slopes (in 10�3 s�1 K�1) l1=T ¼ 3:55�0:32 and l2=T ¼ 1:22� 0:10, and the initial fraction f 1 ¼ 0:6� 0:1.This value for l2=T should be taken as an improvement on the value(0.95) found in Ref. [9]. Using this value we find an improved estimateof the Korringa constant for 8Li in Ag [3]

K ¼ðKcÞ2T1T

1:20� 10�5 sK. (11)

For the high temperature site (site 2), using the demagnetizationcorrected Knight shift from Refs. [9,10], we get K ¼ 1:1� 0:3:

4. Conclusions

We have shown how the Korringa linear temperature depen-dence of the spin lattice relaxation rate is modified by a sitechange transition of the implanted 8Liþ, and developed a simplemodel of this phenomenon that represents some of the behavior.However, the assumption of a single activation law for the sitechange is too simple to model the detailed temperaturedependence. Similar behavior has been observed in other simplemetals, such as Au [3] , Nb [4] and Cu [11]. We also showed howthe baseline from resonance experiments can be used as ameasure of 1=T1.

Acknowledgments

This research is supported by the Center for Materials andMolecular Research at TRIUMF, the Natural Science and Engineer-ing Research Council of Canada (NSERC) and the CanadianInstitute for Advanced Research (CIAR). We also thank RahimAbasalti, Bassam Hitti and Donald Arseneau for technical support.

References

[1] J. Korringa, Physica XVI (1950) 601;C.P. Slichter, Principles of Magnetic Resonance, Springer, Berlin, 1990.

[2] T.J. Parolin, et al., Phys. Rev. Lett. 98 (2007) 047601.[3] T.J. Parolin, et al., Phys. Rev. B 77 (2008) 214107.[4] T.J. Parolin, et al., to be published.[5] L.C. Hebel, C.P. Slichter, Phys. Rev. 113 (1959) 1504;

W.A. MacFarlane, et al., Phys. Rev. B 58 (1998) 1004;J.F. Annett, Physica C 317–318 (1999) 1.

[6] D. Fink, K. Tjan, L. Wang, Rad. Eff. Defects in Sol. 114 (1990) 21.[7] P. Ehrhart, H. Schultz, in: H. Ullmaier (Ed.), Landolt-Bornstein Atomic Defects

in Metals, New Series, Group III vol. 25: Condensed Matter Springer,Heidelberg, 1991 (Chapter 2).

[8] K.H. Chow, et al., Physica B 340–342 (2003) 1151.[9] G.D. Morris, et al., Phys. Rev. Lett. 93 (2004) 157601.

[10] M. Xu, et al., J. Magn. Reson. 191 (2008) 47.[11] Z. Salman, et al., Phys. Rev. B 75 (2007) 073405.