Table Of ContentCrystal field states of Tb3+ in the pyrochlore spin liquid Tb Ti O from neutron
2 2 7
spectroscopy
A. J. Princep,1,∗ H. C. Walker,2 D. T. Adroja,2 D. Prabhakaran,1 and A. T. Boothroyd1,†
1Department of Physics, University of Oxford, Clarendon Laboratory,
Parks Road, Oxford, OX1 3PU, United Kingdom
2ISIS Facility, Rutherford Appleton Laboratory, STFC,
Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
We report time-of-flight neutron scattering measurements of the magnetic spectrum of Tb3+ in
5 Tb2Ti2O7. The data, which extend up to 120meV and have calibrated intensity, enable us to
1 consolidate and extend previous studies of the single-ion crystal field spectrum. We successfully
0 refine a model for the crystal field potential in Tb2Ti2O7 without relying on data from other rare
2 earth titanate pyrochlores, and we confirm that the ground state is a non-Kramers doublet with
predominantly 4 components. Wecompare themodel critically with earlier models.
n |± i
a
J PACSnumbers: 75.10.Kt,75.40.Gb,71.70.Ch,78.70.Nx
0
2
I. INTRODUCTION 13 states of the groundstate 7F6 manifold, whereas oth-
ersincludeallthestatesinthef8 configurationofTb3+.
]
l The main difficulty, however, has been that up to now
e Among the many magnetically frustrated pyrochlore
- only four transitions within the J = 6 level have been
oxides, Tb Ti O (TTO) stands out because of its in-
r 2 2 7 observed unambiguously. Information from other heavy
t triguing low temperature state which is thought to be a
s rare-earth titanates has generally been used to augment
. type of spin liquid.1 TTO shows no sign of any conven-
t the data on TTO so that the six free parameters needed
a tional symmetry-breaking transition (magnetic or struc-
m tural) down to temperatures as low as 50mK (Refs. 2 to describe the single-ion crystal-field Hamiltonian can
be determined independently.
- and 3) despite an antiferromagnetic Curie–Weiss tem-
d In a very recent neutron scattering study,20 a split-
perature of −19K (Ref. 4) and predictions of magnetic
n ting of one of the peaks was resolved and assumed to
orderat1–2K(Refs.5and6). Thereare,however,strong
o originatefromtwo distinct crystalfield transitions. This
short-rangeantiferromagneticcorrelationsinTTOatlow
c
[ temperatures.7–10 ledtoasignificantlydifferentcrystalfieldpotentialcom-
pared with previous models for TTO. For instance, the
The spin liquid state in TTO is not fully understood,
1 wave functions of the ground and first excited doublets
v but a key factor is the low energy part of the crystal- areinterchangedcomparedwithmodelsinformedbydata
7 field-split f electron manifold of Tb3+, which comprises
from other rare-earth titanates. The correctness of this
2 two doublets separated by about 1.5meV (Refs. 2 and
assumption has recently been questioned, and the split-
9 4). This splitting is comparable with the exchange and
4 ting attributed instead to coupling to a phonon18.
dipolar coupling strengths in TTO. Therefore, although
0 Thepurposeofthispaperistoresolvethesediscrepan-
the single-ion ground state is Ising-like, transverse fluc-
. ciesasfaraspossible. Wereportneutronscatteringmea-
1 tuations of the ground state moment appear once inter-
0 surements of TTO which extend to higher energies than
actions are taken into account, and the cooperative na-
5 before. We have observed a crystal field transition that
ture of these fluctuations could lead to a quantum spin
1 wasnotpreviouslydetected,andwehavedeterminedthe
ice state.11,12 Moreover, the two doublets are also con-
: transitionintensities on an absolute scale. By fitting the
v nected by quadrupolarinteractions,allowingcoupling to
i data from TTO alone, using a model that includes in-
X phonons. Indeed,the recentobservationofhybridization
termediate coupling basis states and partial J-mixing,
betweenanacousticphononandacrystalfieldexcitation
r we find a crystal field potential that provides a good de-
a in TTO demonstrates the importance of magnetoelastic
scriptionof the experimentaldata. The modeldescribed
interactions in this system.9,13 So the conditions exist in
in Ref. 20 cannot be reconciled with our data.
TTO forenhancedquantumfluctuations andsuppressed
magneticorderingviabothmagneticandmagnetoelastic
interactions,andanaccuratedeterminationofthecrystal
field states is required for quantitative modelling. II. EXPERIMENTAL DETAILS
Several models for the crystal field potential in TTO
have been reported based on analyses of neutron and A powder sample of mass 16g was prepared by stan-
optical spectra combined with thermal and magnetic dard solid state synthesis. The sample was found to be
data.4,7,14–20 Unfortunately,therearesignificantdiscrep- single phase and of high quality to within the precision
ancies between the published sets of crystal field param- oflaboratoryX-raydiffraction. Neutroninelasticscatter-
eters. These are partly caused by the fact that some of ingmeasurementswereperformedontheMERLINtime-
the models employa truncatedbasis containingonly the of-flight spectrometer at the ISIS Facility.21 The sample
2
15
(a) T = 7K
)
1
−
u.
f. hφi=10◦
1
− 10 hφi=25◦
V
me hφi=40◦
1
−
r
s
b
m
5
(
)
E
φ,
(
S
Figure1. (Coloronline)Neutronscatteringspectrumofpoly- 0
20 40 60 80 100 120
crystallineTb2Ti2O7atT =7K,measuredonMERLINwith
Ei =150meV.Thecolors represent thevalueof E S(Q,E) Energy (meV)
in unitsof mb sr−1f.u.−1 ×
. (b) hφi = 10◦
25
)
1
−
was contained in an aluminium foil packet in the form 90 K
u.
of an annulus of diameter 40mm and height 45mm and f. 20 7 K
sealed in an aluminium can containing helium exchange 1
−
gas. The can was cooled by a closed-cycle refrigerator. V
Spectra were recorded for approximately 4 hours each e
m
with neutrons of incident energy Ei = 65 and 150meV 1 15
at temperatures of T = 7 and 90K. The raw data were −
r
corrected for detector efficiency, sample attenuation and s
b
time-independent background following standard proce- m
dures. Vanadium spectra recorded at the same two in- ( 10
)
cident energies were used to determine the energy reso- E
lution and to convert the intensities into units of cross φ,
section, mbsr−1meV−1f.u.−1, where f.u. stands for the S( 5
formula unit of Tb Ti O .
2 2 7
20 30 40 50
III. RESULTS Energy (meV)
Figure 1 provides an overview of the data recorded at Figure 2. (Color online) Neutron spectrum S(φ,E) of poly-
T = 7K with Ei = 150meV. The spectrum is presented crystalline Tb2Ti2O7 for fixed scattering angle φ. The corre-
intheformofacolormapofE×S(Q,E),whereS(Q,E) spondingQisgivenby¯h2Q2/2mn =Ei+Ef 2(EiEf)12 cosφ,
−
is the scattering intensity as a function of energy E and whereEf =Ei Eandmnistheneutronrestmass. (a)Spec-
−
the magnitude of the scattering vector Q = |Q|. Multi- trumatthreedifferentaveragescatteringanglesrecordedwith
plicationby energysuppressesthe strongelastic andlow Ei=150meV at T =7K,showing magnetic peaksat 49and
61meV and phonon peaks at 72, 91 and 99meV. (b) Com-
energy inelastic scattering and makes the weaker signals
parison of the low angle ( φ =10◦) spectrum at T =7 and
at higher energies more visible. h i
90K, recorded with Ei=65meV.
Previous neutron measurements of the spectrum of
TTO revealed magnetic transitions centred at 1.5, 10,
16 and 49meV. In Ref. 20, a weak feature near 70meV
was also attributed to a magnetic transition, and the confirmed as magnetic by the characteristic reduction of
16meV signal was found to be split into two peaks sep- its intensity with Q due to the magnetic form factor.
arated by 2.5meV consistent with earlier neutron14 and Above50meVthereisaweakfeaturenear61meVwhich
Raman22 spectra. The 10–16meV transitions are clearly decreases with Q, and there are peaks near 72 and 90–
visibleinFig.1althoughtheynotresolvedinthispartic- 100meVwhichincreasewithQconsistentwithscattering
ulardataset.23 The49meVtransitionisalsopresentand from phonons.
3
The part of the spectrum with E > 20meV is rep-
TableI.Observedandcalculatedcrystal-fieldtransitionener-
resented in more quantitative detail in Fig. 2. Fig-
ure 2(a) shows the intensity as a function of energy for giesandintegratedintensitiesfromthespectrumofTb2Ti2O7
measuredat7K.Numbersinparenthesesareexperimentaler-
threedifferentaveragescatteringanglesφtakenfromthe
rorsinthelastdigit. Observedintensitieshavebeencorrected
Ei =150meV run at 7K. The large peak at 49meV and forthedipole form factor ofTb3+ sothat bothIobs andIcalc
the small peak at 61meV both decrease with increasing areforzeroQ. TheIcalc valuesincludetransitionsfromboth
φ (i.e. increasing Q), whereas the peaks at 72, 91 and 0.0 and 1.4meV levels with their respective thermal popu-
99meV all increase with angle. This confirms that the lations at 7K. The best-fit crystal-field parameters used for
latterthreepeaksarecausedbyscatteringfromphonons, the calculations are: B02 = 55.3, B04 = 370.4, B34 = 128.0,
towithinthesensitivityofthemeasurement,whereasthe B06 =114.3, B36 =−114.3, B66 =120.3meV.
49 and 61meV peaks are from magnetic transitions.
Level Eobs Ecalc Iobs(Q=0) Icalc(Q=0)
Figure 2(b) compares φ = 10◦ spectra at 7 and 90K (meV) (meV) (mbsr−1f.u.−1) (mbsr−1f.u.−1)
measured with Ei =65meV. Two features stand out: Γ+3 0.0 0.0 − 2918
(1) The 49meV peak in the 7K spectrum is broader Γ+ 1.4(4)a 1.5 2744
3 −
thantheresolution,withawidth(fullwidthathalfmax- Γ+ 10.2(4) 10.3 1740(60) 1836
2
imum)of4.9meVascomparedwiththeinstrumentalres- Γ+ 16.0(5)b 16.1 990(100) 1010
olutionofabout1.5meVatthis energy. The peak is also 1
Γ+ 39(2) 39.0 31
asymmetric, with a low energy tail that is not wholly 3 ⌉
Γ+ 48(1) 48.2 | 230(35) 62
accounted for by the asymmetry of the resolution func- 2
tion. Onwarmingto90K,thepeakincreasesinintensity Γ+1 49(1) 48.8 ⌋ 184
and shifts down in energy to 47.5meV. Transitions from Γ+ 61(1) 60.8 44(6) 52
3
thethermallypopulated1.5meVfirstexcitedcrystalfield Γ+ 71.0 8
1 − −
level account for some of the peak width at 7K but not
all, and there are no sharp phonon peaks at this energy a Thefirstexcitedstateisdispersiveatlowtemperaturewitha
bandwidthapproaching1meV.2,9,14,20,25
(seeFig.1). Theseobservationssuggestthatthereexists
b Comprisestwopeaksseparatedby2.5meV.14,20
anadditionalcrystalfield levelatabout48meV whichis
strongly connected to the 1.5meV level. The peak posi-
tionsandwidthsat7Kand90Kwouldthenbeexplained
bytransitionsfromthegroundstateand1.5meVlevelto accordingtoanempiricalimplementationoftheanalytic
levels at 48 and 49meV. line shape which contains a contribution from the veloc-
ityselectionviathechopper,24 andanadditionalcompo-
(2) In the range 20–45meV there is considerable addi-
nentdue to the pulse width. The nonmagnetic(phonon)
tionalscatteringat90Krelativeto7K.The90Kscatter-
background was estimated from the high angle part of
ingdoesnotdisplayanyparticularlyprominentfeatures,
the spectrum, which was scaled to match the high en-
butasmallpeakhasgrownatabout24meVandthereis
ergy transfer part of the low angle spectrum. A list of
aweakshouldernear29meVwhichcoincideswithamin-
observedenergylevelsandtransitionintensities at7Kis
imum in the 7K spectrum. These temperature-induced
given in Table I. For ease of comparison, the intensities
featuresarenotaccountedforbythethermalpopulation
have been extrapolated to zero Q via the Q dependence
of phonons (the temperature factor for phonon scatter-
ing[1−exp(−E/k T)]−1increasesbyonly4%at24meV of the magnetic dipole form factor of Tb3+.
B
on warming from 7 to 90K). The additional scattering The crystal field at the Tb3+ site in TTO has point
must therefore derive from transitions out of thermally symmetry3m(D )andisdescribedbytheHamiltonian
3d
excitedmagneticlevels,andsincetheseparationbetween
thepeaksnear16and49meVissignificantlygreaterthan
H =B2C2+B4C4+B4(C4 −C4)+B6C6
25meV there must exist a hitherto undetected level be- CF 0 0 0 0 3 −3 3 0 0
tween these two. Indeed, thermally excited transitions +B6(C6 −C6)+B6(C6 +C6), (1)
3 −3 3 6 −6 6
from the 10 and 16meV levels to a level near 39meV
would give rise to enhanced intensity around 24 and
29meV, as observed. where Bqk are the crystal field parameters and Cqk are
In short,we haveobserveda magnetic peak at61meV Wybourne tensor operators.26 Diagonalisation of HCF
which corresponds to a crystal field level not found in wasperformedin the intermediate coupling scheme with
previous studies, andwe haveindirect evidence for addi- the program SPECTRE.27 To speed up the calculation,
tional levels near 39meV and 48meV. the complete basisofthe f8 configurationofTb3+ (3003
states) was truncated to the lowest 110 states (the com-
ToconstrainthecrystalfieldmodelforTTOastightly
plete 7F term, plus the states 5D , 5D , 5G , 5L and
as possible we shall also consider the transition intensi- 4 3 6 10
5G )extendingto3.5eVabovethegroundstate. Thedi-
ties. Wedeterminedtheintegratedintensitiesofthemag- 5
agonalisationofH includedJ-mixingwithin thetrun-
neticpeaksinthelowanglespectrumat7Kbyfittingan CF
cated basis.
asymmetric pseudo-Voigt line shape to the peaks. The
parametersofthepseudo-Voigtfunctionweredetermined The neutron spectrum of single-ion magnetic transi-
4
tions is given by28
TableII.ComparisonofcrystalfieldparametersBk (inmeV)
q
S(Q,E)= γr0 2e−2W p for Tb2Ti2O7 from different analyses. The parameters are
(cid:16) 2 (cid:17) X iX defined for the Hamiltonian in Eq. (1). The effective Hamil-
i j tonian in terms of Stevens operators Oq,31 which applies for
k
×|hΓ |M (Q)|Γ i|2δ(E −E −E), (2) abasiscomprisingasingleJ levelintheLS couplingscheme,
j ⊥ i j i
where (γr0/2)2 = 72.7mb. The first summation is over iwshHicChFfo=rPTbk3,q+θk(JDkq=O6kq). Tarheeθθ2k=arer1e/d9u9c,eθd4m=at2r/ix16e3le3m5eanntds
the initial states Γi with thermal population pi, and the θ6 = −1/891891. The parameter−s Dkq and Bqk are related
secondsummationisoverthefinalstatesΓj. TheDebye– by Dkq = λqkBqk, where λ02 = 1/2, λ04 = 1/8, λ34 = √35/2,
Waller factor e−2W is taken to be unity at the low tem- λ0 =1/16, λ3 =√105/8 and λ6=√231/16.
6 6 6
peratures of the measurements. We assume the dipole
approximation,inwhichcaseM (Q)canbereplacedby Ref. B02 B04 B34 B06 B36 B66
⊥
4 53.6 318 146 149 143 67.6
−f(Q)gJJ⊥ where f(Q) is the dipole form factor, gJ is 14 60.9 291 103 96.6 −59.9 97.5
the Landé g-factor and J⊥ is the component of the to- 18a 56.0 329 95 107 −77.4 109
tal angular momentum perpendicular to Q. Calculated −
19 67.3 320 119 113 90.5 101
intensities are powder-averagedfor comparison with the −
20 144 268 162 171 349 799
data. 29b 75.3 329 100 111 79.6 130
ThesixcrystalfieldparametersinEq.(1)wererefined This work 55.3 370 128 114 −114 120
by a weighted least-squaresfitting algorithmagainst the −
experimental data given in Table I, with the intensities a ThemodelinRef.18isarefinementofthemodelspresentedin
expressed relative to the 10.2meV peak. The squares Refs.15–17
of the experimental uncertainties were used as recipro- b ParametersforHo2Ti2O7 scaledwiththepoint-charge
relation31 Bqk(Tb)/Bqk(Ho)=hrkiTb/hrkiHo,wherehrkiisthe
cal weights. The crystal field parameters determined for
kth radialmomentofthe4f electrondistribution.
Ho Ti O (Ref. 29) wereusedasstarting parametersfor
2 2 7
the fit. The procedure convergedto an excellent fit with
χ2 =0.95,whereχ2isthestandardnormalisedgoodness- ofHamiltonian(1)]inTableII.Ourparametershavesim-
of-fit statistic. Fits were also performed with different ilar magnitudes and signs to those obtained for TTO in
sets of starting parameters but no other acceptable dis- Refs.4and14–19(whichhoweverreliedonspectroscopic
tinct solutions were found. In particular, fits starting data on other rareearth titanates). The crystalfield pa-
fromtheparametersfoundbyZhanget al.20 didnotcon- rameters are also similar to those of Ho2Ti2O7 (Ref. 29)
verge. and Pr2Sn2O7 (Ref. 30) after taking into account the
The best-fit parameters are given in Table I together differencesintheradialmomentsoftherespective4f or-
with the calculated energy levels and intensities at 7K. bitals. Thelackofsignificantvariationshowninthecrys-
The calculated values agree very well with the obser- tal field potential across different systems implies that
vations, including the absolute intensities which have the local structure and bonding is similar for different
a systematic error of 5–10% from uncertainties in the pyrochlore oxides.
vanadium calibration and in the corrections for attenua- As deduced many years ago,4 the low energy part of
tion and the Q dependence of the magnetic form factor. thespectrumatlowtemperatureiscomposedoftwonon-
Figure 3 compares the predictions of the best-fit model Kramers doublets separated by approximately 1.5meV.
withthespectrameasuredat7Kand90K.Pseudo-Voigt From our model, the largest components of the ground
line shapes have been used to model the resolutionfunc- state and first excited doublet wave functions are found
tion. Theintensitiesarecalculatedinabsoluteunitsfrom to be
Eq. (2) and have not been scaled to fit the data. Over- Γ+(0meV)= 0.968|7F ,±4i∓0.089|7F ,±1i
3 6 6
all, the agreement is very good. The main discrepancy
+0.110|7F ,∓2i±0.181|7F ,∓5i
is that the measured spectrum at 90K is less structured 6 6
above 20meV than the calculated spectrum, which sug- −0.083|7F ,±4i
4
gests that the crystal field levels in this energy range
broaden significantly with temperature. Γ+3(1.5meV)= 0.952|7F6,±5i∓0.176|7F6,±2i
+0.061|7F ,∓1i±0.194|7F ,∓4i
6 6
±0.134|7F ,±5i.
IV. DISCUSSION 5
Theupperandlowersignsgivethetwocomponentsofthe
Wehavepresentedhereasingle-ionmodelbasedsolely doublet, and |2S+1L ,m i identifies the spectroscopic
J J
on measurements on Tb Ti O that successfully de- term and m value for each component. The dominant
2 2 7 J
scribes the general features of the observed magnetic components in the wave functions are from the Hund’s
spectrum of Tb3+ in Tb Ti O . The crystal field pa- rule ground state term |7F i, as expected, but there is
2 2 7 6
rametersofourmodelarecomparedwithpreviouslypub- a non-negligible admixture of the |7F i and |7F i states
5 4
lishedsets[convertedtotheWybournetensorparameters which in the free ion lie 250 and 410meV, respectively,
5
Figure 3. (Color online) Comparison of the observed and calculated spectrum of polycrystalline Tb2Ti2O7. The heavy solid
(red) lines are calculated from Eq. (2) with eigenstates derived from the crystal field parameters given in Table I, and the
individualtransitionsare shown asthin(blue)lines. Thebroken(blue)lines aretheestimated non-magneticbackground,and
thefitted elastic peaks centred on E =0 are shown as pale (green) lines.
above the ground state. For comparison, with the same strongtransitionfrom the firstexcited doublet to a level
crystal field model but working in the pure |7F i basis near72meV.Thezero-Qcrosssectionforthistransition,
6
withtheStevensoperatorformofH insteadofEq.(1) which should appear as a thermally excited peak near
CF
we find that the overall splitting increases from 71meV 70meV, is about 270mbsr−1f.u.−1 at T = 90K, incon-
to 81meV. A list of the wave functions for all the levels sistent with the 90K spectrum shown in the lower right
within the |7F i manifold is given in the Appendix. panel of Fig. 3. Finally, the significant intensity from
6
thermally excited transitions observed between 20 and
Our results are consistent with most previous stud-
40meV — see Fig. 2(b) — is not reproduced.
ies, but are in contrast to the work of Zhang et al.20
who found essentially the same two lowest doublets as Thesuccessofourmodel,whichassumesthatthesplit
above but in the reverse order, i.e. with |7F ,±5i as the peakat16meVcorrespondstoasinglecrystalfieldlevel,
6
dominant components of the ground state doublet. This raisesquestions about the nature of this excitation. One
difference arises because Zhang et al. assumed in their possibility is thatthe splitting could be the resultof dis-
analysis that the splitting of the 16meV peak was due order, e.g. Tb/Ti site mixing, which could produce two
to two distinct crystal field transitions. Although the slightlydifferentlocalenvironmentsfortheTbsite. How-
model of Zhang et al. fits their own data well, our mea- ever,the factthatno splitting is detectable inanyofthe
surements reveal a number of problems with it. Firstly, othertransitions,especiallytheΓ doubletswhosedegen-
3
the magnetic peak we observe at 61meV was not iden- eracy is lifted once the 3m symmetry is broken, suggests
tified as a crystal field transition by Zhang et al., and that there is only one Tb3+ environment. Further, the
therefore not used to constrain the model. Second, their splittingistoolargetobeexplainedbythesametwo-ion
modelpredicts alevelat101meV witha verylargecross magnetic coupling model which accounts for the disper-
section for transitions to it from the ground state dou- sionofthefirstexciteddoublet.6 KlekovkinaandMalkin
blet: 440mbsr−1f.u.−1 at zero Q. This prediction is have suggested that the peak splitting is caused by cou-
inconsistent with our data since such an intense transi- pling to a phonon with an energy near 16meV at the
tion wouldproduce a peak centred at101meV in Figs. 1 zone centre.18 Indeed, hybridization of acoustic phonons
and 2(a) which, after correction for the Tb3+ form fac- with the 1.5meV and 10meV crystal field levels has re-
tor, would be of roughly the same size as the peak at cently beenobservedin neutronspectra ofTTO,9,13 and
49meV. Third, the model of Zhang et al. predicts a strongthermally-inducedphononanomaliesareobserved
6
Table III. Eigenfunctions of the best-fit single-ion crystal field model for Tb3+ in Tb2Ti2O7. The crystal field Hamiltonian is
given in Eq. (1), and the values of thebest-fit crystal field parameters are given in the caption of Table I. The list below is of
the energy levels within the 7F6 manifold and gives only the coefficients of mJ belonging to 7F6 . A blank entry means a
| i | i | i
zero coefficient.
Energy (meV)
mJ 0.0 0.0 1.5 1.5 10.3 16.1 39.0 39.0 48.2 48.8 60.8 60.8 71.0
6 0.145 0.179 0.687 0.678 0.033
− − −
5 0.181 0.952 0.182 0.033
− −
4 0.968 0.194 0.071 0.096
− − −
3 0.686 0.675 0.134 0.173 0.066
− − − − −
2 0.110 0.176 0.956 0.057
−
1 0.089 0.061 0.058 0.974
− −
0 0.105 0.019 0.982
−
+1 0.089 0.061 0.058 0.974
−
+2 0.110 0.176 0.956 0.057
−
+3 0.686 0.675 0.134 0.173 0.066
−
+4 0.968 0.194 0.071 0.096
− −
+5 0.181 0.952 0.182 0.033
− −
+6 0.145 0.179 0.687 0.678 0.033
− − − −
and attributed to magneto-phononcoupling.32 The exis- sured by neutron spectroscopy. The model is in very
tence of magnetoelastic modes was suggested as an ex- goodquantitativeagreementwiththeexperimentalspec-
planation for why frustration is not relieved in TTO by trum,including theabsoluteintensity,andconfirmsthat
any conventional symmetry-breaking transitions at low the single-ion magnetic properties at low temperature
temperatures.9,13 Although our data cannot shed much are controlled by two non-Kramers doublets separated
light on this interesting question, we mention that the byabout1.5meV.The wavefunctionsofthesestatesare
crystal field states in TTO do have strong quadrupole approximately|m i=|±4iand|±5i,respectively. Our
J
momentssoaperturbationtothesingle-ionstatesinvolv- analysis is more tightly constrained by experiment than
ing a coupling via orbitals is plausible. Measurements hashithertobeenpossible,andresolvesanuncertaintyin
on single crystals are needed to identify any coupling to theliteratureaboutthecompositionofthegroundstate.
specific phononmodes andto see whether the size ofthe The splitting of the 16meV peak remains unexplained,
splitting varies throughout the Brillouin zone. and needs further investigation.
AsfarasthephysicalpropertiesofTTOareconcerned,
the most important result from this study is that the
ground state is dominated by the components |7F ,±4i.
6
This means that at temperatures where the first excited
ACKNOWLEDGMENTS
doublet has negligible population TTO has strong Ising-
like anisotropy with the moments confined to the quan-
tization axis, i.e. to the local h111i directions. For the We are grateful to Michel Gingras and Bruce Gaulin
ground state doublet, we calculate that the components for helpful comments. This work was supported by the
of the zero-field spectroscopic g-tensor parallel and per- UK Engineering & Physical Sciences Research Council.
pendicular to the quantization axis are g = 10.7 and
k
g = 0 (g = 0.07 in a field of 1 Tesla), consistent with
⊥ ⊥
previous estimates.19
Appendix A: Single-ion eigenfunctions of Tb Ti O
V. CONCLUSION 2 2 7
We have successfully refined a model for the crystal The eigenfunctions for the best-fit model are given in
field in Tb Ti O against the magnetic spectrum mea- Table III.
2 2 7
∗ [email protected] 1 J. S. Gardner, M. J. P. Gingras, and J. E. Greedan, Rev.
† [email protected] Mod. Phys.82, 53 (2010).
7
2 J.S.Gardner,S.R.Dunsiger,B.D.Gaulin,M.J.P.Gin- 17 V. V. Klekovkina, A. R. Zakirov, B. Z. Malkin, and L. A.
gras, J. E. Greedan, R. F. Kiefl, M. D. Lumsden, W. A. Kasatkina, J. Phys.: Conf. Ser. 324, 012036 (2011).
MacFarlane, N. P. Raju, J. E. Sonier, I. Swainson, and Z. 18 V. V. Klekovkina and B. Z. Malkin, Optics and Spec-
Tun, Phys.Rev.Lett. 82, 1012 (1999). troscopy 116, 849 (2014).
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