^{27}Al Impurity-Satellite NMR and Non-Fermi-Liquid Behavior in U_{1-x}Th_xPd_2Al_3

10 Pages·0.51 MB·English

by  Chia-Ying Liu

27Al Impurity-Satellite NMR and Non-Fermi-Liquid Behavior
in U Th Pd Al
1−x x 2 3
Chia-Ying Liu and D. E. MacLaughlin
Department of Physics, University of California, Riverside, California 92521
H. G. Lukefahr, G. Miller, and K. Hui
Whittier College, Whittier, California 90608
0
0
M. B. Maple, M. C. de Andrade, and E. J. Freeman
0
2 University of California, San Diego, California 92093-0319

n
(January 7, 2000)
a
J
7 Non-Fermi-liquid(NFL)behaviorinthef-sublattice-dilutedalloysystemU1−xThxPd2Al3hasbeen
studied using 27Al nuclear magnetic resonance (NMR). Impurity satellites due to specific U near-
] neighbor configurations to 27Al sites are clearly resolved in both random and field-aligned powder
l
e samples. The particular configuration associated with each satellite is identified by comparison of
- calculated and observed satellite intensities. The spatial mean K and rms spread δK of impurity
r
t satellite shifts, which arerelated tothemean χand rmsspread δχ of theinhomogeneous suscepti-
s bility, have been measured in field-aligned powders with the crystalline c axis both perpendicular
.
t andparallel to theexternal field. Satellites corresponding to only oneuranium nearneighbor were
a
m chosen for analysis, since in this case δχ(T)/χ(T) = δK/K independent of the (unknown) spa-
tial correlation of the random susceptibility inhomogeneity. The relatively narrow lines observed
-
d atlow temperaturessuggest thatdisorder-inducedinhomogeneityof thef-ion–conduction-electron
n hybridization is not the cause of NFL behavior in these alloys: at low temperatures the experi-
o mental values of δχ(T)/χ(T) are much smaller than required by disorder-driven models. This is
c in contrast to results in at least some alloys with disordered non-f-ion nearest neighbors to f ions
[ (“liganddisorder”),wheredisorder-driventheoriesgivegoodaccountsofNFLbehavior. Ourresults

suggest that f-ion dilution does not produce as much inhomogeneity of the hybridization strength
1
v as substitution on ligand sites.
8
PACS numbers: 71.27.+a, 75.30.Mb, 76.60.Cq.
7
0
1
0 I. INTRODUCTION behavior have emerged: (a) proximity to a zero-
0 temperature quantum critical point (QCP), of either a
0 The magnetism of heavy-fermion f-electron materials single-ion or a cooperative nature,2,3 and (b) the ef-
/
t is quenched at low temperatures by conduction-electron fect of lattice disorder on the Kondo properties of the
ma (Kondo)screening. Amany-bodygroundstateisformed f ions.2,4–9 In the QCP picture the NFL behavior is
that has traditionally been described within Landau’s due to quantum fluctuations associated with a critical
d- Fermi-liquid theory both for dilute alloys1 (the single- point at zero temperature. This mechanism is operative
n impurity Kondo problem) and for concentrated “Kondo in uniform systems, since disorder is not required. In
o lattice” systems (with or without lattice disorder). But disorder-driven scenarios the effect of structural disor-
c thethermodynamicandtransportpropertiesofanumber der on f-electron many-body effects such as the Kondo
v: of f-electron “heavy-fermion” metals and alloys do not effect andthe Ruderman-Kittel-Kasuya-Yosida(RKKY)
i behaveaspredictedbyFermi-liquidtheory.2Theinappli- interaction between f ions produces a broad inhomoge-
X
cability of this picture is signaledby (a) weak power-law neous distribution of the local susceptibilities χj associ-
ar or logarithmic divergences of the specific heat Sommer- ated with the f ions. Uncompensated Kondo ions far
feldcoefficientγ(T)=C(T)/T andthemagneticsuscep- from the singlet ground state, which are not described
tibility χ(T), both of which are constant in Fermi-liquid byFermi-liquidtheory,giverisetolargevaluesofχj and
theory, and (b) a temperature dependence of the electri- the NFL properties of the material. QCP and disorder-
cal resistivity which is weaker than the T2 prediction of drivenmechanismsneednotbemutually exclusive,how-
Fermi-liquidtheory. Better theoreticalandexperimental ever, since critical fluctuations of a disordered system
understanding ofthese so-callednon-Fermi-liquid(NFL) might also be involved in NFL behavior.
systems has been the goal of a considerable amount of The single-ion “Kondo disorder” picture,4–6 in which
researchin recent years. theuncompensatedionsareassumednottointeract,was
Two broad classes of theoretical explanation of NFL developedtoexplainnuclearmagneticresonance(NMR)
1
shift K of the field for resonance at fixed frequency;14,15
thisshiftisexpectedtobeproportionaltotheU-ionsus-
ceptibility χ. Any inhomogeneity in χ leads to a corre-
U(cid:13) Th(cid:13) Pd(cid:13) Al(cid:13) T(cid:13) f(cid:13)rom C(cid:13) (cid:13)((cid:13)T)(cid:13)(cid:13) sponding distribution of shifts that broadens the NMR
)(cid:13)K 40(cid:13) 1-(cid:13)x(cid:13) x(cid:13) 2(cid:13) 3(cid:13) TK(cid:13) (cid:13)f(cid:13)rom r(cid:13)(cid:13) ((cid:13)T)(cid:13)(cid:13) line. Therelationbetweenthermsspreadδχinχandthe
( e TK(cid:13)K (cid:13)(cid:13)f(cid:13)rom c(cid:13)(cid:13) ((cid:13)T)(cid:13)(cid:13) NMRlinewidthσisdescribedelsewhere,5,16,17whereitis
ru shown that σ/(KH0), where K is the spatially averaged
tar relative shift and H0 is the applied field, is an estima-
e 20(cid:13)
p tor for the fractional rms spread δχ/χ, where χ is the
me TN(cid:13) (cid:13) spatiallyaveragedsusceptibility. We canwriteσ/(KH0)
T AFM(cid:13) Kondo + NFL(cid:13) equivalently as δK/K, where δK ≡ σ/H0 is the relative
T(cid:13)
c(cid:13) SC(cid:13) rms spread in shifts.
0(cid:13) NMR line broadening can also arise from dynamic
0.0(cid:13) 0.2(cid:13) 0.4(cid:13) 0.6(cid:13) 0.8(cid:13) 1.0(cid:13)
(lifetime) effects due to nuclear spin-lattice relaxation.
Th concentration (cid:13)x(cid:13) PulsedNMRtechniquescanbeusedtoestimatesuchlife-
time broadening independently of the spectral width,14
FIG.1. PhasediagramofU1−xThxPd2Al3. Antiferromag- and we have found that in U1−xThxPd2Al3 spin-lattice
netic (AFM), Superconducting(SC), and Kondo/non-Fermi- relaxation is far too small to contribute significantly to
liquid (Kondo + NFL) regions are shown. NFL behavior oc- the observed spectral linewidths.
curs for x >∼ 0.6, where the Kondo temperature TK was de- In a diluted magnetic alloy such as U1−xThxPd2Al3
terminedfrom measurementsofspecificheatC(T),electrical the NMR shift is also distributed because of the spatial
resistivityρ(T),andmagneticsusceptibilityχ(T). Datafrom dependence of the RKKY interaction and the random
Ref. 11. positions of the magnetic ions, even if the susceptibility
associatedwiththeseionswereuniform. Thusadditional
broadening due to susceptibility inhomogeneity may be
experiments in the NFL alloys UCu5−xPds, x=1.0 and difficult to resolve. This is unlike the situation in an al-
1.5, that revealed wide distributions of frequency shifts
loywithliganddisorder (i.e.,disorderinthenonmagnetic
reflecting the required susceptibility inhomogeneity. In
ions of the compound), where the magnetic-ion sublat-
theKondodisordermodelstructuraldisordergivesriseto
ticeisordered. Inthiscasetheonlysourceofbroadening
a wide distribution of local Kondo temperatures (T ) ;
K j is susceptibility inhomogeneity,as longas the liganddis-
this leads to a distribution of χ which becomes corre-
j orderdoesnotaffecttheRKKYinteractionsignificantly.
spondingly wide at low temperatures. The “Griffiths-
phase”modelofCastroNeto et al.7–9 takesintoaccount Even if it does the two broadening mechanisms can be
distinguished.17
RKKY interactions between uncompensated f-ion mo-
If, however, specific near-neighbor magnetic-ion con-
ments in the disordered system. These RKKY interac-
figurations of the observed nuclei are probable enough
tions couple the uncompensated moments into clusters,
in a dilute alloy, and if the shifted NMR frequencies
the thermal behavior of which canbe describedin terms
of these nuclei are large enough to be resolved, rather
of Griffiths singularities10 associated with the distribu-
thanmerelycontributingto“dilutionbroadening”ofthe
tion of cluster sizes.
resonance line, then the shifts and linewidths of these
Both the Kondo-disorder and Griffiths-phase theories
impurity satellites may be studied separately.18 Impu-
makedefinitepredictionsoftheinhomogeneousspreadin
rity satellites provide a much better characterization of
magneticsusceptibilitythatiscapableofbeingmeasured
the inhomogeneoussusceptibility distributioninanalloy
by magnetic resonance techniques such as NMR. The
presentpapercomparesthesepredictionswith27AlNMR with f-ion sublattice dilution (such as U1−xThxPd2Al3)
than is possible if the satellites are not resolved.
spectra inunalignedandfield-alignedpowder samplesof
An important result of the present studies was
the NFL alloys U1−xThxPd2Al3, x=0.7, 0.8, and 0.9. the observation of resolved 27Al impurity satellites in
hibTithseNiFsoLstrbuechtauvriaolr afollroyintseerrmieesdiUat1e−xtTohhxiPghd2tAhlo3riuexm- U1−xThxPd2Al3. The shifts and linewidths of these
satellites have been analyzed to provide information on
concentrations.11 ThephasediagramofU1−xThxPd2Al3 the inhomogeneous distribution of local susceptibility in
is shown in Fig. 1. The heavy-fermion end compound
this system. We find that the width of the susceptibility
UPd2Al3 exhibits coexistence of antiferromagnetic order
distribution is considerably smaller than required to ex-
(T = 14 K) and superconductivity12 (T = 2 K) that
N c plain NFL behavior, and a different mechanism must be
persistsforlowThconcentrations(x<0.2). Forx>0.6
sought. Thisresultsuggeststhat, incontrasttothesitu-
the electrical resistivity, specific heat, and magnetic sus-
ationinligand-disorderalloys,f-sublattice disordermay
ceptibility are all indicative of NFL behavior.13
not produce enough inhomogeneity in the f-electron–
The RKKY coupling between 27Al nuclei and U-ion
conduction-electron hybridization to drive the NFL be-
spinsaffectsthe27AlNMRinanumberofways,ofwhich
havior.
themostimportantforourpurposesistheparamagnetic
2
II. DISORDER-DRIVEN NFL THEORIES
(cid:13))
U
In the following we briefly describe the single-ion e 80(cid:13)
Kondo-disorder4–6 and Griffiths-phase7–9 models, after lo
m U(cid:13) Th(cid:13) Pd(cid:13) Al(cid:13)
which we consider their applicability to the NFL behav- /u 60(cid:13) 0.1(cid:13) 0.9(cid:13) 2(cid:13) 3(cid:13)
m
ior in U1−xThxPd2Al3. e
The simplest implementation of the single-ion Kondo (cid:13)3
–
disorder model assumes that the f ions are coupled to 0(cid:13)1 40(cid:13)
(
the conduction-electron bath by a random distribution y c(cid:13)
of Kondo coupling constants g = ρJ, where ρ is the tilib 20(cid:13) ab(cid:13)
conduction-electrondensityofstatesattheFermienergy itp c(cid:13) c(cid:13)
e
andJ isthelocal-moment–conduction-electronexchange c
s 0(cid:13)
energy. Then g is related to the Kondo temperature T u
K S 0(cid:13) 100(cid:13) 200(cid:13) 300(cid:13)
by
Temperature (K)(cid:13)
T =E exp(−1/g), (1)
K F
FIG.2. Temperaturedependenceoftheanisotropicsuscep-
where E is the Fermi energy of the host metal. Thus a
F tibility in U0.1Th0.9Pd2Al3. Circles: ab-plane susceptibility
modestdistributionofg cangiverisetoabroaddistribu-
χab(T). Triangles: c-axis susceptibility χc(T). Curve: fit of
tionofT ifmostvaluesofgaresmall. Ifthedistribution
K Griffiths-phase model (Refs. 7–9) to χab(T).
function P(T ) is broad enough so that P(T =0) does
K K
not vanish, then at any nonzero temperature T those f
ions for which TK <T will not be compensated. Fermi- picture. The procedure used to compare the Griffiths-
liquid theory does not apply to them, and they give rise phasetheorywithexperimentissimilartothatdescribed
to the NFL behavior. above for the Kondo-disorder analysis: bulk susceptibil-
Such uncompensated f ions dominate thermal and ity data are fit to the corresponding theoretical expres-
transport properties at low temperatures. The magnetic sions, and the calculated δχ(T)/χ(T) is compared with
susceptibility is correspondingly distributed, as can be NMR data.
seen from the Curie-Weiss law
C
χ(T,TK)= (2) III. EXPERIMENT
T +αT
K
(α ∼ 1) that approximately characterizes the Kondo Samples of U1−xThxPd2Al3, x = 0.7, 0.8, and 0.9,
physics, and is inhomogeneous on an atomic scale. were prepared as described previously.11,13 The arc-
Fitting this model to the temperature dependence of meltedingotswerecrushedandpassedthrougha100-µm
the bulk (i.e., spatially averaged) magnetic susceptibil- sieve. 27Al NMR experiments were carried out on un-
ity χ(T) gives the distribution of g, which is then used aligned powders and also on epoxy-cast powder samples
to predict δχ(T)/χ(T).4,5 inwhichthesingle-crystalpowdergrainswerealignedby
In the Griffiths-phase theory various physical proper- a 6-T magnetic field during hardening ofthe epoxy. The
tiesarepredictedtodivergeatlowtemperaturesasweak crystal symmetry is hexagonal (space group P6/mmm)
power laws of temperature. For example, the electronic andthus the susceptibility is uniaxially anisotropic. The
specific heat C(T) and the spatially averaged magnetic directionoflargestmagneticsusceptibilityisinthebasal
(ab) plane, so that in order to orient the c axes of the
susceptibility χ(T) are given by
grains it was necessary to rotate the sample around an
C(T)/T ∝χ(T)∝T−1+λ, (3) axis perpendicular to the magnetic field while the epoxy
hardened.19 Therotationaxisthendefinesthec-axisori-
and the fractional rms spread δχ(T)/χ(T) is given by entationof the grains. The anisotropicsusceptibility ob-
tained from a field-aligned sample of U0.1Th0.9Pd2Al3 is
δχ(T) ∝T−λ/2. (4) showninFig.2,wherethestronganisotropythatpermits
χ(T) the field alignment can be seen.
AlsoshowninFig.2istheresultoffittingtheGriffiths-
The nonuniversal exponent λ is a parameter that deter- phase model prediction for the uniform susceptibility
minesthedegreeofNFLcharacter. TheGriffithsphaseis to the experimental χ (T) data, as described above in
ab
characterizedbyλ≤1,sothatthesusceptibilitydiverges Sec. II. Aside from an overallscale factor the fit param-
atzerotemperature(thecaseλ=1ismarginalandgives eters are the exponent λ and a high-energy cutoff ǫ0 for
risetologarithmicdivergences). SinceC(T)/T andχ(T) the distribution of cluster energies. Best fit was found
have the same temperature dependence the Wilson ra- for λ = 0.90 and ǫ0/kB = 130 K. A good fit was also
tio χ(T)T/C(T) is independent of temperature in this
3
U(cid:13) Th(cid:13) Pd(cid:13) Al(cid:13)
0.1(cid:13) 0.9(cid:13) 2(cid:13) 3(cid:13)
U(cid:13) Th(cid:13) Pd(cid:13) Al(cid:13) (cid:13) 0.8(cid:13)
0.2(cid:13) 0.8(cid:13) 2(cid:13) 3(cid:13)
Central Transition(cid:13) (a) Minority lines(cid:13)
130 K(cid:13)
0.6(cid:13)
56.5 MHz(cid:13)
Extra lines(cid:13) )(cid:13)%
QSautaedllrituepso(cid:13)le(cid:13) QSautaedllrituepso(cid:13)le(cid:13) ( tfih 0.4(cid:13)
S
R 0.2(cid:13)
M
N
lA 0.0(cid:13)
7(cid:13) 0.2(cid:13)
2 (b) Principal Lines(cid:13)
50.5(cid:13) 51.0(cid:13) 51.5(cid:13)
0.0(cid:13)
Magnetic Field (kOe)(cid:13)
0(cid:13) 100(cid:13) 200(cid:13)
FIG. 3. Representative 27Al quadrupole-split NMR spec-
Temperature (K)(cid:13)
trum from an unaligned powder sample of U0.2Th0.8Pd2Al3.
Thespectrum is characterized bya narrow centraltransition
and broad powder-pattern quadrupole satellites (Ref. 14). FIG. 4. Temperature dependence of 27Al NMR shifts in
Extra lines are visible on the low-field side of the central unaligned powder samples of U1−xThxPd2Al3, x = 0.7
transition, and to some extent on the low-field sides of the (squares),0.8(triangles),and0.9(circles). (a)Minoritylines.
quadrupolesatellites. (b) Principal lines.
obtained with the Kondo-disorder model, where the pa- shifted to first order in the coupling constant e2qQ and
rameters describing the disorder were taken to be the depend on crystallite orientation, so that the quadrupo-
Fermi energy EF [cf. Eq. (1)] and the mean g and stan- larsatellitesare“powder-patternbroadened”bytheran-
dard deviation δg of an assumed Gaussian distribution domorientationsofthepowdergrainsinthesample. The
ofcoupling constants. The values EF =1 eV, g =0.149, centraltransitionisshiftedonlytosecondorderine2qQ,
and δg = 0.020 gave the best fit (not shown). The re- however, and suffers much less powder-pattern broaden-
sults ofthese fits wereusedtocalculatenumericalvalues ingthanthesatelliteswhene2qQissmallerthanthe27Al
of δχ(T)/χ(T), which are compared with results of the nuclear Zeeman frequency.
NMR experiments in Sec. IV. The most important feature of Fig. 3 is the group of
Field-swept 27Al NMR spectra were obtained using extra lines on the low-field side of the central transi-
pulsed-NMR spin-echo signals and the frequency-shifted tion. (There is also a hint of this structure in some of
and summed Fourier transform processing technique de- the quadrupole satellites.) The extra lines, which also
scribed by Clark et al.20 The echo-decay lifetime T2 was appear for Th concentrations x = 0.7 and 0.9, were
found to be sufficiently long (hundreds of µs) so that no initially suspected to be due to spurious metallurgical
correction for the echo decay was needed. phases, althoughx-ray powder diffraction measurements
indicatedthesamplesweresingle-phase.13Morethanone
extra “minority” line was observed, but in the following
A. Spurious phases or impurity satellites in onlythe mostintenseminorityline is comparedwiththe
U1−xThxPd2Al3? “principal” central transition (Fig. 3).
The temperature dependence of the principal and mi-
Figure 3 shows as an example a spectrum from nority shifts is shown in Fig. 4. It can be seen that
an unaligned powder sample of U0.2Th0.8Pd2Al3. In the shift of the principal line is independent of temper-
U1−xThxPd2Al3themmmpointsymmetryoftheAlsite ature, whereas the minority line exhibits a Curie-Weiss-
is lowerthan cubic, so that the 27Al nuclear Zeeman lev- like temperature-dependent shift. If the minority line
els are split by the quadrupole interaction between the were due to a spurious phase, then these results suggest
27Al nuclear quadrupole moment Q and the crystalline that the principal phase is a nearly pure thorium com-
electric field gradient q.14 Then quadrupole satellite res- pound,whereasthespuriousphaseorphaseshaveahigh
onances, corresponding to mI ↔ mI−1 (mI 6= 1/2) nu- uraniumconcentration. Thisseemsratherunlikely,given
clear spin transitions, are observed in the form of broad the absence of x-ray evidence for phase segregation and
peaks shifted from a narrow central (1/2↔−1/2) tran- the chemical similarity of thorium and uranium, and we
sition. The positions of the quadrupole satellites are taketheresultsshowninFig.4asinitialevidenceagainst
4
0.8(cid:13) U(cid:13) Th(cid:13) Pd(cid:13) Al(cid:13) H(cid:13) (cid:13) || c(cid:13) (cid:13)
0.1(cid:13) 0.9(cid:13) 2(cid:13) 3(cid:13) 0(cid:13)
U(cid:13) Th(cid:13) Pd(cid:13) Al(cid:13)
)(cid:13) 0.1(cid:13) 0.9(cid:13) 2(cid:13) 3(cid:13)
% 0.6(cid:13) Central Transitions(cid:13)
( 4.2 K (cid:13)
tfihS 0.4(cid:13) 73 MHz(cid:13) Bulk Line(cid:13)
R
M
Quadrupole(cid:13)
N 0.2(cid:13)
(cid:13)7 lA QSautaedllrituepso(cid:13)le(cid:13) Impurity(cid:13) Satellites(cid:13)
2 0.0(cid:13) Satellites(cid:13)
0(cid:13) 10(cid:13) 20(cid:13) 30(cid:13) 40(cid:13) 50(cid:13)
Susceptibility (10(cid:13)–3(cid:13) emu mole(cid:13)–1(cid:13))(cid:13)
64(cid:13) 65(cid:13) 66(cid:13) 67(cid:13)
FIG.5. Clogston-Jaccarino plot of 27Al principal- and mi-
Magnetic Field (kOe)(cid:13)
nority-line NMR shifts vs. bulk (spatially averaged) sus-
ceptibility in unaligned powder samples of U1−xThxPd2Al3,
x= 0.7 (squares), 0.8 (triangles), and 0.9 (circles). Temper- FIG. 6. Representative 27Al NMR field-aligned spectrum
atureis an implicit parameter. Theminority-line data(filled from a field-aligned powder sample of U0.1Th0.9Pd2Al3 for
symbols) show similar behavior for different Th concentra- H0 k c. The bulk NMR lines and impurity satellites (indi-
tions x. Extrapolations to zero susceptibility (high tempera- cated for thecentral transition only) are grouped into a cen-
ture)agreewiththetemperature-independentprincipalshifts tral transition and four quadrupole satellites. The impurity
(open symbols at zero susceptibility). satellites have slightly different anisotropic NMR shifts and
quadrupole splittings than the bulk lines. The sharp lines
indicate good alignment of the powder-grain c axes.
the spurious-phase hypothesis.
Figure 5 gives a Clogston-Jaccarino plot of shift ver-
sus susceptibility per mole uranium, with temperature the temperature-independent principal-line shifts. This
an implicit variable. A linear relation between shift and agreementwouldbeacompletecoincidenceiftheminor-
χ(T)is expectedfor impurity satellites if the transferred ity lines were due to spurious phases, but follows natu-
hyperfinefieldthatcouplesthenuclearspintotheUmo- rally if they are impurity satellites.
mentistemperature-independent,andisindeedobserved We conclude that the observed properties of the spec-
forsmalltomoderatevaluesofχ(T). Theslopesofthese tra establish the minority lines as impurity satellites
linear relations are more or less independent of thorium rather than due to spurious phases.
concentrationx,asexpectedforalocalhyperfineinterac-
tion. [The nonlinear behavior for larger χ(T) is not well
understood, but may arise from changes in crystal-field B. Spectra from field-aligned samples
level populations with temperature.]
The temperature independence of the principal-line The 27Al impurity-satellite spectra from unaligned
shiftisexplainedbythelargenumberofconfigurationsof powder samples of U1−xThxPd2Al3 described above are
moredistantUionsandtheoscillatorydependenceofthe difficultto interpret,due to possiblepreferentialorienta-
RKKY interaction, which lead to broadening but little tion and broadening from shift anisotropy. We therefore
average shift contribution to the principal line.21 Vary- usedfield-alignedsamplesforfurther NMR experiments.
ing the Th concentration x can change the host-metal Figure6 showsa representativefield-alignedspectrum
band structure and lead to an x-dependent host Knight fromU0.1Th0.9Pd2Al3 forappliedfieldH0 paralleltothe
shift. The fact that the principal-line shifts do not vary c axis(H0 kc). The sharplines indicate goodalignment
monotonically with Th concentration is not well under- of the powder-grain c axes, and the impurity satellites
stood, but may be due to shift anisotropy together with aremoreapparentthanintheunalignedpowderspectra.
sample-dependent preferential orientation of the powder The lines in the center of the spectrum are the (1/2 ↔
grains. −1/2)centraltransitionbulklineandimpuritysatellites.
Comparison of the principal- and minority-line shifts Impurity satellites can also be seen associatedwith each
yields strong additional evidence that the minority lines quadrupole satellite. These are well resolvedon the low-
areimpuritysatellitesratherthanduetospuriousphases. fieldsideofthequadrupole-splitspectrumbutnotonthe
One of the most striking features of Fig. 5 is the high-field side, due to the combination of an anisotropic
fact that extrapolations of the minority-line shifts to NMR shift and slightly larger quadrupole splittings (at
zero χ(T) (infinite temperature) are in agreement with
5
fixed frequency) of the impurity satellites relative to the
bulk lines.
Itcanalsobe seeninFig.6that the quadrupolesatel-
lites are somewhat broader than the central transition,
due presumably to incomplete alignment and/or disor-
der in the quadrupole interaction; as mentioned above
this results in first-order quadrupole broadening of the
quadrupole satellites but only second-order broadening
of the central transitions. Henceforth we consider only
the central-transition bulk line and impurity satellites.
Field-swept spectra were taken with a field range of
about 300 Oe around the central transition and a small
fieldstep(∼3Oe)tohavegoodresolutionofnarrowlines.
Spectra were obtained as a function of temperature and
field for both field directions (H0 k c and H0 ⊥ c) on
field-aligned samples of U1−xThxPd2Al3, x = 0.7,0.8,
and 0.9.
Impurity satellites and line shapes in dilute magnetic
alloys have been treated by Walstedt and Walker,21
FIG. 7. Crystal structure of U1−xThxPd2Al3. The num-
who showed on very general grounds that in the ab-
bers on the (U,Th) sites indicate the nearest, next-nearest,
senceofsusceptibilityinhomogeneitythelineshapesand and third nearest shells to the reference 27Al site (X). The
widths of the bulk line and impurity satellites are the latter is at the center of the rectangle formed by the four
same. Inhomogeneity results in an increase of the satel- nearest-neighbor (U,Th) sites.
lite linewidths relative to that of the bulk line. We find
that a Lorentzian line shape fits the bulk line best. Fol-
lowing the result of Walstedt and Walker, the impurity Thisimpliesthataneffectivenext-nearest-neighborshell
satellites were fit with a Lorentzianwith the same width isacombinationofthesecondandthirdshellsinthecrys-
as the bulk line, convoluted with an extra (Gaussian) tal structure. The distance from the reference Al site to
broadening function that describes the susceptibility in- thenearestshellis3.402˚A,tothesecondshellis5.096˚A,
homogeneity. andtothe thirdshellis 6.828˚A,sothatthe differencein
Simplestatisticalconsiderationswereusedtoconstrain (U,Th)-Al distances between the second and third shells
the fits. The probability Pn0(x) of finding n and only n isabout1.7˚A.Thisisnotsmallontheatomicscale,and
n
uraniumimpuritiesinagivennear-neighbor(Th,U)shell wedonotunderstandwhytheRKKYcouplingconstants
around an Al site is given by for these shells are apparently so nearly equal.
For these choices of near-neighbor shell sizes, and
Pn0(x)=yn(1−y)n0−nCn0, (5) assuming an isotropic RKKY interaction,22 we have
n n
5 distinct nearest-neighbor-shell configurations (nnn =
where y = 1−x is the U concentration, n0 is the total 0,1,2,3,4) and 9 distinct next-nearest-neighbor-shell
numberofUsitesintheshell,andCnn0 =n0!/n!(n0−n)! configurations (nnnn = 0,1,2,...,8). The total num-
isthebinomialcoefficient. Theintensityofeachline(i.e., ber of distinct configurations associated with these two
theareaundertheline)includingthebulkline(forwhich shells is therefore 5×9=45. Table I gives the probabil-
n=0forallresolvednear-neighborshells)isproportional ities of the six most probable of these 45 configurations
to this probability. The fitting procedure then consists fromEq.(5)fortwouraniumconcentrationsy =0.1and
of taking the number of most probable U-ion configura- 0.2 (x = 0.9 and 0.8, respectively). The configurations
tions to be the number of resolved impurity satellites, are designated by (p,q), where p and q are the number
fixing the ratios of the satellite and bulk line intensities of U ions in the nearest-neighbor and (effective) next-
according to Eq. (5), and varying the shifts and widths nearest-neighbor shells, respectively. We have taken the
of all lines for best fit. fit spectrum to include the lines corresponding to these
The crystal structure of U1−xThxPd2Al3, including six configurations (i.e., five impurity satellites and the
near-neighbor (Th,U) shells around a reference 27Al site bulkline)asshowninFig.8(a). Fixingtherelativeareas
out to the third shell, is shown in Figure 7. Each of ofthelinesaccordingtoTableIleavesasfreeparameters
the first three near-neighbor shells contains four sites the area of the bulk line and the shifts and linewidths of
(n0 = 4). We found, however, that if we took n0 = 4 all the lines.
for both the nearest-neighbor and next-nearest-neighbor The positions and widths of the lines in Fig. 8(a) are
shells we could not fit the spectra well. Among vari- the result of a fit to the spectrum of U0.1Th0.9Pd2Al3
ous possibilities we find that the choice of nn0n = 4 for for H0 k c, T = 30 K, and a spectrometer frequency of
the nearest-neighbor shell and nn0nn = 8 for the next- 17.221 MHz. This fit is shown together with the data in
nearest-neighbor shell gives the best fit to the spectra. Fig. 8(b). It is important to note that the assignment
6
Configuration x=0.9 x=0.8
(0,0) 0.2825 0.0687 U(cid:13) Th(cid:13) Pd(cid:13) Al(cid:13) H(cid:13) (cid:13) || c(cid:13) (cid:13)
0.1(cid:13) 0.9(cid:13) 2(cid:13) 3(cid:13) 0(cid:13)
(0,1) 0.2510 0.1374
(1,0) 0.1255 0.0687
(1,1) 0.1116 0.1374 (a) Bulk line & impurity satellites(cid:13)
(0,2) 0.0976 0.1203
(1,2) 0.0434 0.1203 (0,0)(cid:13)
Total 0.9116 0.6528
TABLEI. Probability offindingtheconfiguration (p,q)of (0,1)(cid:13)
U ions in the two nearest-neighbor (Th,U) shells about an (1,1)(cid:13)
(1,2)(cid:13)
Al site in U1−xThxPd2Al3 [from Eq. (5)]. Here p and q are
the number of U ions in the nearest-neighbor and (effective) (0,2)(cid:13) (1,0)(cid:13)
next-nearest-neighborshells, respectively.
(b) Fit to data(cid:13)
30 K (cid:13)
of each line to a configuration is consistent with the os-
17.221 MHz(cid:13)
cillatory RKKY interaction. Consider for example the
(0,1) and (1,0) impurity satellites, which are on oppo-
site sides of the (0,0) bulk line. Then one more uranium
inthesecondshellshouldpushthe(0,2)satellitefurther
away from the (0,0) line than the (0,1) satellite. Simi-
larly, the (1,1) satellite shouldlie between the (1,0) and
(0,1) satellites due to the opposite signs of the nearest-
and next-nearest-shell interactions, and the (1,2) satel-
15.45(cid:13) 15.50(cid:13) 15.55(cid:13)
lite should lie between the (1,0) and (0,2) satellites. All
Magnetic Field (kOe)(cid:13)
these properties are satisfied by the fits without having
beenputin“byhand.” Wearguethatthesuccessofthis
fitting procedure is excellent evidence that the satellites FIG. 8. 27Al NMR spectrum from a field-aligned powder
have been correctly identified. sample of U0.1Th0.9Pd2Al3, H0 k c. (a) Bulk line and five
The model is very successful, in the sense that it gives impurity satellites included to give best fit (see text and Ta-
good fits to the field-aligned spectra for both field di- ble I for nomenclature). (b) Circles: NMR data. Solid line:
rections and for x = 0.9 and 0.8. Shown in Fig. 9 are Fit curve.
some examples of the spectra and their fits. It can be
seen, however, that the fit for x = 0.8 is not quite as
IV. DISORDER AND NFL BEHAVIOR IN
good as for x = 0.9. This might be associated with
the fact that the total probability to find any of the six U1−xThxPd2Al3
resolved-satellite U-ion neighbor configurations (cf. Ta-
ble I) is quite high (0.9116) for x = 0.9, but is only Field-swept 27Al NMR spectra from field-aligned pow-
0.6528 for x = 0.8. The remaining 39 configurations der samples were obtained at a spectrometer frequency
must contribute to the broadening function, and if the of 17.221 MHz over the temperature range 5–250 K.
probability associated with a few of these configurations For H0 k c all impurity satellites have very weak or
is appreciable one might suspect that a simple Gaussian temperature-independent shifts (not shown). Such be-
approximation to the line shape could break down. In havior is expected because the c axis is the magnetic
line with this speculation it was found that spectra from “hard” axis.
U0.3Th0.7Pd2Al3 were even more difficult to fit. Figure 10 gives the temperature dependence of 27Al
Even so, U1−xThxPd2Al3 is the first system in which impurity satellite shifts Kab(T) relative to the bulk line
impurity satellites have been resolved at such high mag- for H0 ⊥ c. There are strong similarities between re-
netic impurity concentrations (y = 0.2). This is due in sults for the two concentrations,and the (0,1)and (0,2)
part to the relatively small number of f-ion sites in the satellites have stronger temperature dependences than
near-neighbor shells compared, for example, to the case therestoftheimpurity satellites. Thelinewidths σab(T)
in dilute CuFe alloys18where nnn =12. of impurity satellites (0,1) and (0,2) also have strong
0
temperature dependence at low temperatures as shown
inFig.11. Itisnotsurprisingtoseethateachsatellitehas
almost the same width at high temperatures where the
magnetic susceptibility is small. The (1,2) satellite for
U0.2Th0.8Pd2Al3 isanexception,asitshowsonlyaweak
temperaturedependence andalargerwidthathightem-
7
U(cid:13) Th(cid:13) Pd(cid:13) Al(cid:13) U(cid:13) Th(cid:13) Pd(cid:13) Al(cid:13) H(cid:13) (cid:13) ^(cid:13)(cid:13) c(cid:13) (cid:13)
1–(cid:13)x(cid:13) x(cid:13) 2(cid:13) 3(cid:13) 1–x(cid:13) x(cid:13) 2(cid:13) 3(cid:13) 0(cid:13)
(a) (cid:13)x(cid:13) = 0.8(cid:13)
H(cid:13) || c(cid:13) (cid:13)
30 K(cid:13) 0(cid:13) 1(cid:13) (a) (cid:13)x(cid:13) = 0.9 17.221 MHz(cid:13) ((01,,10))(cid:13)(cid:13)
18.225 MHz(cid:13) (1,1)(cid:13)
(0,2)(cid:13)
(cid:13)) (1,2)(cid:13)
%
(
tfih 0(cid:13)
S
R
M
16.35(cid:13) 16.40(cid:13) 16.45(cid:13) 16.50(cid:13)
N
e
3(b0) K (cid:13)x(cid:13)(cid:13) = 0.8(cid:13) H(cid:13)0(cid:13) ^(cid:13)(cid:13) c(cid:13) (cid:13) vitale 1(cid:13) (b) (cid:13)x(cid:13) = 0.8 17.221 MHz(cid:13)
17.221 MHz(cid:13) R
lA
(cid:13)7
2
0(cid:13)
(c) (cid:13)x(cid:13) = 0.9(cid:13) H(cid:13) ^(cid:13)(cid:13) c(cid:13) (cid:13) 0(cid:13) 100(cid:13) 200(cid:13)
0(cid:13)
30 K(cid:13) Temperature (K)(cid:13)
17.221 MHz(cid:13)
FIG. 10. Temperature dependence of impurity satellite
shifts Kab(T) relative to the bulk line for H0 ⊥ c in
U1−xThxPd2Al3 at 17.221 MHz. (a) x = 0.9. (b) x = 0.8.
The(0,1)and(0,2)satelliteshavestrongtemperaturedepen-
dences.
15.4(cid:13) 15.5(cid:13) 15.6(cid:13)
Magnetic Field (kOe)(cid:13)
single-U-ion impurity satellite means that no informa-
tion can be obtained concerning ξ (T), but by the same
χ
FIG.9. Representativefitsto27AlNMRfield-alignedspec- token the quantity σ (T)/K (T)H =δK (T)/K (T)
ab ab ab ab
tra from U1−xThxPd2Al3 alloys. (a) x = 0.8, B k c. givesδχ(T)/χ(T)independentofthe(unknown)valueof
(b) x = 0.8, B ⊥ c. (c) x = 0.9, B ⊥ c. Circles: NMR ξ (T).
χ
data. Solid lines: Fit curves. Figure 12 plots δK (T)/K (T) for the (0,1) im-
ab ab
purity satellite versus the bulk susceptibility χ(T) in
peraturescomparedtotheothersatellites. Thisbehavior U0.1Th0.9Pd2Al3, H0 ⊥ c, again with temperature an
implicit parameter. Also shown are the theoretical
is notunderstood, but it canbe seen fromFig. 8(a)that
predictions of δχ(T)/χ(T) from the single-ion Kondo-
the (1,2) satellite is the weakest of the impurity satel-
disorder and Griffiths-phase theories, obtained as de-
lites and is not well resolved; there may be considerable
scribed in Sec. II. In spite of the fact that at low tem-
systematic error in the parameters for this satellite.
peraturesthesatellitesaresignificantlybroaderthanthe
Only the (0,1) and (0,2) impurity satellites have
central transition (Fig. 11), the theoretical predictions
strong temperature-dependent linewidths and shifts for
H0 ⊥ c. We use the (0,1) satellite in U0.1Th0.9Pd2Al3 considerablyoverestimatethe experimentalresults. This
suggests that the effect of disorder in this system is too
forfurtheranalysis,becauseforthissatellitethereisonly
one U moment in the immediate 27Al environment. This weak to account for its NFL behavior. The satellites
are simply too narrow, a fact which, ironically, is essen-
isincontrasttothesituationinligand-disorderNFLsys-
tial to their experimental observation. Impurity satel-
(tRemefss,.e2.g3.a,nUdC1u75−),xPwdhxer(eRetfhse. 5fanidon1s6)aarendcoCnecRenhtRrautSeid2 lites have also been observed in Y0.8Th0.2−yUyPd3,24
but in this case the satellite widths become very wide
and nuclear spins couple strongly to substantially more
and unresolved at low temperatures. We also note
than one neighboring f-ion moment. In the latter case
it can be shown that the correlation length ξ (T) that that in U0.1Th0.9Pd2Al3 δKab(T)/Kab(T) varies rela-
χ
tivelyslowlywithχ(T)(Fig.12),i.e.,theNMRlinewidth
characterizes the random spatial variation of the sus-
ceptibility has to be taken into account.5 The use of a is nearly proportional to the shift. This is in contrast to
8
U(cid:13) Th(cid:13) Pd(cid:13) Al(cid:13) H(cid:13) (cid:13) ^(cid:13)(cid:13) c(cid:13) (cid:13) 1.0(cid:13)
1–x(cid:13) x(cid:13) 2(cid:13) 3(cid:13) 0(cid:13)
U(cid:13) Th(cid:13) Pd(cid:13) Al(cid:13)
0.1(cid:13) 0.9(cid:13) 2(cid:13) 3(cid:13)
0.8(cid:13)
(0,1) impurity satellite(cid:13)
(a) (cid:13)x(cid:13) = 0.9 17.221 MHz(cid:13)
(0,0)(cid:13) c(cid:13)/(cid:13)
20(cid:13) (0,1)(cid:13) c(cid:13) 0.6(cid:13)
((11,,01))(cid:13)(cid:13) d(cid:13) (cid:13) ,b(cid:13)a Griffiths-phase fit(cid:13)
(cid:13)K
10(cid:13) ((01,,22))(cid:13)(cid:13) /(cid:13)(cid:13)Kb(cid:13)a 0.4(cid:13) Kondo-disorder fit(cid:13)
d(cid:13) 0.2(cid:13)
)(cid:13)
e
O
0.0(cid:13)
( 0(cid:13)
h 30(cid:13) 0(cid:13) 20(cid:13) 40(cid:13) 60(cid:13) 80(cid:13)
tdiw (b) (cid:13)x(cid:13) = 0.8 17.221 MHz(cid:13) c(cid:13) (10(cid:13)–3(cid:13) emu/mole U)(cid:13)
ab(cid:13)
e
n
iL 20(cid:13)
FIG. 12. Comparison of fractional rms susceptibility
spread δχ(T)/χ(T) from the Kondo-disorder (solid curve)
10(cid:13) and Griffiths-phase (dashed curve) theories with fractional
27Al NMR width δKab/Kab of the (0,1) impurity satellite
in U0.1Th0.9Pd2Al3, H0 ⊥ c (data points). The predictions
0(cid:13) clearly overestimate the NMR data, indicating that disorder
0(cid:13) 100(cid:13) 200(cid:13) is not themajor source of NFL behavior in this alloy.
Temperature (K)(cid:13)
like temperature-dependent linewidths and shifts for
FIG.11. Bulklineandimpuritysatellitelinewidthsσab(T) H0 ⊥ c. But the linewidths do not increase much more
vs. temperature for H0 ⊥ c in U1−xThxPd2Al3 at 17.221 rapidly with decreasing temperature than the shift, so
MHz. (a) x=0.9. (b) x=0.8. The (0,1) and (0,2) satellite
that δK/K does not exhibit the rapid increase with
linewidths have strong temperature dependences at low T.
bulk susceptibility χ expected from the disorder-driven
Each satellite has almost the same width at high T with the
models.5,7 These mechanisms also overestimate the ob-
exception of the(1,2) satellite for x=0.8.
servedlinewidthatlowtemperatures,suggestingthatthe
disorder in this system is not strong enough to account
thebehaviorexpectedfromboththeKondo-disorderand for its NFL behavior.
Griffiths-phase pictures, where the spread in susceptibil- In the disorder-driven models the origin of the dis-
ities is a rapidly-growing fraction of the average suscep- order is variation of the f-electron/conduction-electron
tibility as the temperature is lowered.4,7 hybridization matrix element with local f-ion environ-
ment. Disorder is found to be an important contribu-
tor to NFL behavior in alloys with ligand disorder, such
V. CONCLUSIONS as UCu5−xPdx (Ref. 5) and CeRhRuSi2 (Ref. 17). One
might suspect, however, that the immediate f-ion en-
27AlNMRinthe U1−xThxPd2Al3 alloysystemhasre- vironment is not as strongly disordered in dilute solid
solutions of f ions, given that in the limit of infinite di-
vealedsatelliteNMRlinesduetospecificuraniumconfig-
lution all f ions have identical environments. NFL be-
urations around Al sites. These impurity satellites facil-
havior in such systems might therefore be due to some
itate determination ofthe effect of disorderon paramag-
other mechanism. This conclusion must be regarded
netism in this system. The probability offinding a given
as speculative, however, since U concentrations of 10%
configuration,whichisrelatedtotheintensityofthecor-
and 20% can hardly be considered dilute. In addition,
responding line, follows a simple statistical calculation.
We used a procedure that fixed the intensities according U1−xThxPd2Al3 istheonlyf-iondilutedsystemstudied
to date using NMR.
to their probabilities to fit the field-aligned spectra for
x = 0.9 and 0.8 and both field directions (H0 k c and NFL behavior in U1−xThxPd2Al3 is observed over a
H0 ⊥ c). Each impurity satellite is thereby associated waniddetrraanngspeoorftTphrocpoenrcteienstroabteioynssi,nagnled-itohnestchaelrimngo.1d1ynTahmesiec
with a specific near-neighbor uranium configuration.
results suggest that a quantum critical point associated
The linewidths and shifts in U1−xThxPd2Al3 do not
have much temperature dependence for H0 k c, as is with cooperative behavior is not the NFL mechanism in
this system. Two models that rely neither on cooper-
also the case for the c-axis susceptibility χ (T). In con-
c
ative effects nor on disorder have been applied specifi-
trast, two of the impurity satellites have Curie-Weiss-
9
cally to U1−xThxPd2Al3. These are (a)the quadrupolar 15This NMR shift is sometimes called the Knight shift, but
Kondo model,11 which assumes a non-Kramers doubly weprefertoreservethelattertermexclusivelyfortheshift
degenerate nonmagnetic ground state, and (b) the elec- dueto conduction electrons in metals.
tronic polaron model of Liu,25 which assumes that the 16O. O. Bernal, D. E. MacLaughlin, A. Amato, R. Feyer-
f-electron energies are close to the Fermi level and that herm, F. N. Gygax, A. Schenck, R. H. Heffner, L. P. Le,
transport involves polaron-like hopping between f sites. G. J. Nieuwenhuys, B. Andraka, H. von L¨ohneysen, O.
Neither of these theories predicts strong disorder in the Stockert,and H. R.Ott, Phys.Rev. B 54, 13000 (1996).
magnetic susceptibility, and from the standpoint of our 17Chia-Ying Liu, D. E. MacLaughlin, A. H. Castro Neto,
H.G.Lukefahr,J.D.Thompson,J.L.Sarrao,andZ.Fisk,
NMR results both remain candidates for NFL behavior
Phys.Rev.B, to bepublished.
in U1−xThxPd2Al3. 18J.B.BoyceandC.P.Slichter,Phys.Rev.B13,379(1976).
19M.Lee,G.F.Moores, Y.Q.Song,W.P.Halperin,W.W.
Kim, and G. R. Stewart, Phys.Rev. B 48, 7392 (1993).
ACKNOWLEDGMENTS
20W.G.Clark, M.E.Hanson, F.Lefloch,andP.Segransan,
Rev.Sci. Instrum.66, 2453 (1995).
We are grateful to A. H. Castro Neto for helpful dis- 21R. E. Walstedt and L. R. Walker, Phys. Rev. B 9, 4857
cussions and comments. This researchwas supported by (1974).
theU.S.NSF,Grantnos.DMR-9418991(U.C.Riverside) 22AnisotropyintheRKKYinteractionwouldfurtherpowder-
andDMR-9705454(U.C. SanDiego),bythe U.C. River- pattern broaden the satellite lines proportionally to the
side Academic Senate Committee on Research, and by susceptibility. We will see in Sec. IV that any such contri-
the Research Corporation (Whittier College). bution to the linewidth does not affect the conclusions of
thispaper.
23T. Graf, J. D. Thompson, M. F. Hundley,R. Movshovich,
Z.Fisk,D.Mandrus,R.A.Fisher,andN.E.Phillips,Phys.
Rev.Lett. 78, 3769 (1997).
24H. G. Lukefahr, E. Camacho, G. Miller, K. Hui, D. E.
MacLaughlin, Chia-Ying Liu, M. B. Maple, and D. G.
1P. Nozi`eres, J. Low Temp. Phys. 17, 31 (1974). Gajewski (unpublished).
2See, for example, articles in Proceedings of the Conference 25S.H. Liu, Physica (Amsterdam) B 240, 49 (1997).
on Non-Fermi Liquid Behavior in Metals, Santa Barbara,
California, 1996, edited by P. Coleman, M. B. Maple, and
A.J. Millis, J. Phys.: Condens. Matter 8 (1996).
3A.M.TsvelikandM.Reizer,Phys.Rev.B48,9887(1993).
4O. O. Bernal, D. E. MacLaughlin, H. G. Lukefahr, and B.
Andraka,Phys.Rev. Lett.75, 2023 (1995).
5D. E. MacLaughlin, O. O. Bernal, and H. G. Lukefahr, J.
Phys.: Condens. Matter 8, 9855 (1996).
6E. Miranda, V. Dobrosavljevi´c, and G. Kotliar, J. Phys.:
Condens. Matter 8, 9871 (1996).
7A.H.CastroNeto,G.Castilla,andB.A.Jones,Phys.Rev.
Lett. 81, 3531 (1998).
8M.C.deAndrade,R.Chau,R.P.Dickey,N.R.Dilley,E.J.
Freeman, D. A. Gajewski, M. B. Maple, R. Movshovich,
A. H. Castro Neto, G. E. Castilla, and B. A. Jones, Phys.
Rev.Lett. 81, 5620 (1998).
9A. H. Castro Neto, G. Castilla, and B. A. Jones (unpub-
lished).
10R.B. Griffiths, Phys. Rev.Lett. 23, 17 (1969).
11M. B. Maple, M. C. de Andrade, J. Herrmann, Y.
Dalichaouch, D. A. Gajewski, C. L. Seaman, R. Chau, R.
Movshovich,M.C.Aronson,andR.Osborn,J.LowTemp.
Phys.99, 223 (1995).
12C. Geibel, C. Schank,S. Thies, H. Kitazawa, C. D. Bredl,
A. Bohm, M. Rau, A. Grauel, R. Caspary, R. Helfrich,
U. Ahlheim, G. Weber, and F. Steglich, Z. Phys. B 84, 1
(1991).
13Y.DalichaouchandM.B.Maple,Physica(Amsterdam)B
199-200, 176 (1994).
14G.C.Carter,L.H.Bennett,andD.J.Kahan,Prog.Mater.
Sci. 20, 1 (1977).
10