“the (low t) magnetic ground state of gd2ti2o7 is not 4k” · alex hannon arwel hughes ann terry...

“The (low T) Magnetic Ground State of Gd2Ti2O7 is

not 4k”Joe Paddison and Ross Stewart

DIVISION HEADSean Langridge

Steve HullGroup Leader

Dave Keen Kevin Knight

Matthias Gutmann

Bill Marshall

Ron Smith

CRYSTALLOGRAPHYLARGE SCALE STRUCTURES

John WebsterGroup Leader

Steve King Richard Heenan

Phil TaylorTim Charlton

Alan Soper

DISORDERED MATERIALS

Daniel BowronGroup Leader

Winfried Kockelmann

Aziz Daoud-AladineMatt Tucker

Alex Hannon

Arwel Hughes Ann Terry

August 2012

Sylvia Imberti

ISIS SCIENCE DIFFRACTION AND MATERIALS DIVISION

Luke Clifton Sarah RogersChristian Kinane

Dmitry Khalyavin Shu Yan Zhang

Max Skoda

Pascal Manuel

Antonella Scherillo

Joseph Kelleher

Rob Dalgliesh

Tristan Youngs

Diego Alba Venero Larmor Vacancy

VacancyPA/Divisional

Secretary

Martin Owen Jones Hazel Sparkes

Genoveva Burca

Saurabh Kabra

ENERGY MATERIALS CO-ORDINATOR

Daniel Myatt

Nina Steinke

Jason Gardner (ANSTO)Andrew Goodwin!(Oxford)

Georg Ehlers (SNS)

Joe Paddison (ISIS, Oxford)Pascal Manuel

(ISIS)

Acknowledgments

Oleg Petrenko (Warwick)

Aziz Daoud-!Aladine (ISIS)

Dmitry Khalyavin!(ISIS)

Gd2Ti2O7 - AF pyrochlore

B. dc and ac magnetic susceptibility measurements

The dc magnetic susceptibility � was measured using asuperconducting quantum interference device magnetometer⌅Quantum Design, San Diego⌃ in the temperature range2–300 K. The ac susceptibility �ac was measured at differentfrequencies by the mutual inductance method. The primarycoil of the mutual inductor is energized by a frequency gen-erator ⌅DS 335, Stanford Research Systems⌃ and the outputacross the two identical secondary coils, wound in oppositedirections, was measured using a lock-in amplifier ⌅SR-830DSP, Stanford Research Systems⌃. The sample susceptibilitywas determined from the difference in the outputs with thesample in the middle of the top secondary coil and withoutthe sample. The cryostat used for the temperature variation isdescribed in the section below.

C. Specific-heat measurements

The specific heat of the sample in the form of a pellet(⌦100 mg) was measured in the temperature range 0.6–35K using a quasiadiabatic calorimeter and a commercial He-liox sorption pumped 3He cryostat supplied by Oxford In-struments. The sample was mounted on a thin sapphire platewith apiezon for better thermal contact. Underneath the sap-phire plate a strain gauge heater and a RuO2 temperaturesensor were attached with G-E varnish. The temperature ofthe calorimetric cell was controlled from the 3He pot on theHeliox. The sample temperature was measured using an LR-700 ac resistance bridge at a frequency of 16 Hz. The spe-cific heat of the sample was obtained by subtracting the con-tribution of the addendum, measured separately, from thetotal measured heat capacity.

III. EXPERIMENTAL RESULTS

The dc susceptibility � Fig. 2⌅a⌃⇥ measured at an appliedfield of 0.01 T vs temperature is found to obey the Curie-Weiss behavior in the range 10–300 K. An effective mag-netic moment of 7.7⇧B /Gd3⇤ obtained from the Curie-Weiss fit is close to the expected value of 7.94⇧B /Gd3⇤ forthe free ion 8S7/2 , and a paramagnetic Curie temperature�CW of �9.6(3) K indicates antiferromagnetic interactionsbetween the Gd3⇤ spins. It is worth noting that � starts de-viating at a temperature of the order of �CW as it ought to befor a ‘‘conventional’’ system undergoing a transition to long-range order. That �CW is predominantly due to exchangeinteractions as opposed to crystal-field effects is confirmedby measurements on the magnetically diluted system(Gd0.02Y0.98)2Ti2O7, for which �CW is much reduced and ofthe order of ⇤�0.9 K Fig. 2⌅b⌃⇥. The absence of any mag-netic ordering down to 2 K in the concentrated system, eventhough �CW is about five times larger than this temperature,suggests the presence of important magnetic frustration in-hibiting the occurrence of magnetic long-range order.In search of a possible magnetic ordering below 2 K, ac

susceptibility �ac was measured down to 0.3 K. The tempera-ture variation of �ac for different frequencies ⌅Fig. 3⌃ exhibitstwo features, a broad peak centered at about 2 K and a sharpdown turn below about 1 K, the latter possibly signaling atransition to long-range antiferromagnetic order. �ac(⌥) ap-pears to be independent of frequency that would seem to rule

out a spin-glass state, as opposed to what has been found inother pyrochlore oxides such as Y2Mo2O7,22–24Tb2Mo2O7,22,25 and the frustrated Gd3Ga5O12 garnet.9,10,12The specific heat Cp as a function of temperature is

shown in Fig. 4. There is a broad peak centered around 2 K

FIG. 2. ⌅a⌃ Inverse molar susceptibility 1/� of Gd2Ti2O7 againsttemperature in the temperature range T⇥2–25 K, and in the tem-perature range T⇥2–300 K in the inset. ⌅b⌃ Inverse molar suscep-tibility 1/� of (Gd0.02Y0.98)2Ti2O7 against temperature in the tem-perature range T⇥2–25 K, and in the temperature range T⇥2–300 K in the inset.

PRB 59 14 491TRANSITION TO LONG-RANGE MAGNETIC ORDER IN . . .

N P Raju, et. al., Phys. Rev B 59 (1999) 14489

AF interactions, θCW = ~ -10 K !Leading terms in the Hamiltonian expected to be Heis. exch. (~5 K NN) + dipole-dipole (~0.8 K NN)

AF Pyrochlore - Gd2Ti2O7

Long-range order achieved at low-temperatures in Gd2Ti2O7 There are two transitionsTN

1 ~1 K TN

2 ~0.7 K

P Bonville, et. al. J. Phys.: Condens. Matter 15 7777 (2003) O A Petrenko, et. al. Phys. Rev. B 70 012402 (2004)

Entropy not fully recoveredabove 5K - short-range order

R ln(8)

Proposed structureMean-field model (NN exchange): Soft-modes along (hhh) directions may provide order-by disorder - entropic selection of ground-state due to density of soft modes (Raju, 1999) !!4th-order terms in Free energy select a k=0 AF ground state: Palmer-Chalker State (PC) - energetically, not entropically selected (Palmer, 2000) !Mossbauer: Gd moments lie perp. to <111> (Bonville, 2003) !!

“The magnetic structure determination for Gd2Ti2O7 and Gd2Sn2O7 is currently a mess: The neutron scatterers predict bizarre states.....” (Brammall,et. al., arXiv:1008.1203v1, 2010)

addition of further near neighbour exchange leads to other possible ordering wavevectors (Wills, 2006)

Gd2Sn2O7 not so “bizarre”L40 Letter to the Editor

Figure 3. The magnetic structure basis vectors labelled according to the different irreducible co-representations for Gd2Sn2O7.

diffraction and prevents contributions of the individual basis vectors from being refined. Forthis reason only ψ6 was used in the refinement and the final fit is presented in figure 2.While the value of the ordered moment, 6(1) µB/Gd3+, obtained by scaling the magneticand nuclear peaks is imprecise due to the uncertainty over the isotopic composition of the Gdand the concomitant neutron absorption, it is consistent with the free-ion value (7 µB) and thatmeasured by Mossbauer spectroscopy [13].

Realization of the PC state in Gd2Sn2O7, but not in Gd2Ti2O7, indicates that the magneticHamiltonian of the titanate contains additional terms. Cepas and Shastry [19] have suggestedthat next-neighbour exchange may stabilize magnetic ordering at k = ( 1

212

12 ), though the

corresponding region in the parameter space was found to be tiny. Also, possible exchangepaths were not investigated in their work as both types of third-neighbour exchange (figure 1)were assumed to be equal.

The pyrochlore A2B2O7 structure has two inequivalent oxygen sites: O1 at (x, 18 , 1

8 )

and O2 at ( 38 , 3

8 , 38 ). The oxygen parameter is x = 0.335 and 0.326 for the stannate and

the titanate, respectively [20]. Using this information we have determined that the nearest-neighbour gadolinium ions are connected with short Gd–O1(2)–Gd bonds. The second-neighbour exchange J2 is produced by two distinct Gd–O1–O1–Gd bridges. The O1–O1distance in the first path is 2.63 A with two equal bond angles of 118◦. In the secondpath, |O1–O1| = 3.04 A, the angles are 148◦ and 98◦. (Distances and angles are given forGd2Ti2O7.)

There are two types of third-neighbour pairs in the pyrochlore lattice indicated in figure 1as J31 and J32, which correspond to three- and two-step Manhattan (city-block) distances,respectively. The superexchange J31 is determined by two equivalent Gd–O1–O1–Gd pathswith |O1–O1| = 3.04 A and two equal angles of 148◦. The superexchange J32 is produced by

• Larger lattice constant • Gd-O-Gd bond angles

less favourable for further NN exchange

• PC state is found by neutron diffraction

Letter to the Editor L39

13 20 27 34 41 48 55 62 69 76 83

-4000

4000

Inte

nsity

(a.u

.)

60000

52000

44000

36000

28000

20000

12000

-12000

-20000

2 θ (°)

Figure 2. Fit to the magnetic diffraction pattern of Gd2Sn2O7 obtained from the ψ6 basis stateby Rietveld refinement. The dots correspond to experimental data obtained by subtraction of thatmeasured in the paramagnetic phase (1.4 K) from that in the magnetically ordered phase (0.1 K). Thesolid line corresponds to the theoretical prediction and the line below to the difference. Positionsfor the magnetic reflections are indicated by vertical markers.

The magnetic diffraction peaks, figure 2, can be indexed with the propagation vectork = (0 0 0). The different types of magnetic structure can be classified in terms of theirreducible co-representations of the reducible magnetic co-representation, c"mag. In the caseof Gd2Sn2O7 (space group Fd 3m with k = (0 0 0) and a Gd3+ ion at the 16d crystallographicposition) this can be written as: c"mag = 1c"3+ + 1c"5+ + 1c"7+ + 2c"9+ [17]6. Theirassociated basis vectors are represented in figure 3. Inspection reveals that c"3+ correspondsto the antiferromagnetic structure observed in FeF3 [18], c"5+ to the linear combinationobserved in the model XY pyrochlore antiferromagnet Er2Ti2O7 [11], c"7+ to the manifoldof states proposed as the ground states for the Heisenberg pyrochlore antiferromagnet withdipolar terms (the PC ground state) and c"9+ to a spin-ice like manifold observed in thenon-collinear ferromagnetic pyrochlores such as Dy2Ti2O7 [3]. While the phase transitionin Gd2Sn2O7 has been shown to be first order, which allows ordering according to severalirreducible co-representations, it is commonly found that the terms which drive the transitionto being first order are relatively weak and cause only minor perturbation to the resultantmagnetic structure. Following this, we examined whether the models detailed above couldfit the observed magnetic neutron diffraction spectrum. The goodness of fit parameter, χ2,for the fit to models characterized by each irreducible co-representation are: c"3+ (69.0),c"5+ (35.6), c"7+ (5.18), c"9+ (13.6). We find that the magnetic scattering can only bewell modelled by c"7+, the PC state in which the moments of a given Gd tetrahedron areparallel to the tetrahedron’s edges. In this state each moment is fixed to be perpendicular tothe local three-fold axis of each tetrahedron, consistent with Mossbauer data [13, 14]. Powderaveraging leads to the structures ascribed to ψ4, ψ5 and ψ6 being indistinguishable by neutron

6 The co-representations are real and are labelled according to the notation of Kovalev [17] for the parentrepresentation and whether the antiunitary halfing group was created according to d(a) = ±δ(aa−1

0 )β, where d(a) is

the matrix representative of the antiunitary symmetry element a, δ(aa−10 ) is the matrix representative of the unitary

symmetry element aa−10 , a0 is an antiunitary generating element and β is an unitary matrix.

A S Wills, et al.. J. Phys.: Condens. Matter 18, L37–L42 (2006).

D20, ILL

Gd2Sn2O7 not so “bizarre”

WISH data on isotopic sample

PC state confirmed at zero-field

WISH, ISIS

but recently..

Ord

ered

Gd

Mom

ent

0

1

2

3

4

5

6

7

8

Field (T)0 2 4 6 8 10 12 14

• Only 3.5 μB ordered moment inPC state at B = 0 T

• Diffuse scattering • partial order

WISH, ISIS

Possible interactionsL38 Letter to the Editor

J32

J31

J2

Figure 1. Pyrochlore lattice of vertex-sharing tetrahedra. Next-neighbour exchanges are shown bylong-dashed lines.

transition temperature, which are described by propagation vectors [hhh] [8]. Later, Palmerand Chalker showed that quartic terms in the free energy lift this degeneracy and stabilize afour-sublattice state with the ordering vector k = (0 0 0) (the PC state) [9].

Among various pyrochlore materials Gd2Ti2O7 and Gd2Sn2O7 are believed to be goodrealizations of Heisenberg antiferromagnets. Indeed, the Gd3+ ion has a half-filled 4f-shell withnominally no orbital moment. However, a strong intrashell spin–orbit coupling mixes 8S7/2 and6P7/2 states, leading to a sizable crystal-field splitting. Recent electron spin resonance (ESR)measurements on dilute systems gave comparable ratios of the single-ion anisotropy constantD > 0 to the nearest-neighbour exchange J for the two compounds: D/J ∼ 0.7 [10]. Thiscorresponds to a planar anisotropy for the ground state. Magnetic properties of the stannate andthe titanate would therefore be expected to be very similar. In this light, the contrasts betweenthe low-temperature behaviour of Gd2Sn2O7 and that of the analogous titanate are remarkable.While the titanate displays two magnetic transitions, at ∼0.7 and 1 K, to structures with theordering vector k =

! 12

12

12

"[11, 12], the stannate undergoes a first-order transition into an

ordered state near 1 K [13]. Furthermore, Mossbauer measurements indicate that in the latterthe correlated Gd3+ moments still fluctuate as T → 0 K [14].

In this letter we demonstrate that Gd2Sn2O7 orders with the PC ground state, evidencefor an experimental realization of a Heisenberg pyrochlore antiferromagnet with dipolarinteractions. We also note that the magnetic structure observed in Gd2Sn2O7 differs from thoseobserved in the closely related Gd2Ti2O7, indicating that an extra interaction is at play in thelatter which leads to the observed differences.

In order to reduce the absorption of neutrons, a 500 mg sample of Gd2Sn2O7 enriched with160Gd was prepared following the conditions given in [13]. Powder neutron diffraction spectrawere collected with neutrons of wavelength 2.4 A using the D20 diffractometer of the InstituteLaue-Langevin, Grenoble at two temperatures below TN = 1 K (0.1 and 0.7 K), as well as onetemperature in the paramagnetic phase above TN (1.4 K). Due to the high residual absorptionof the Gd sample, extended counting times of 5 h per temperature were required. The magneticcontribution to diffraction at 0.1 K could be well isolated from scattering by the cryostat wallsand dilution insert by subtraction of the spectrum at 1.4 K. Symmetry calculations were madeusing SARAh [15] and Rietveld refinement of the diffraction data using Fullprof [16] togetherwith SARAh.


-0.5 0.0 0.5-0.5

0.0

0.5

(0 0 0)

(q q 0)

(q 0 0)

(q q’ 0)

1 2_ ( _ _ 1 1

2 2 )

J2

J31

Figure 4. Instability wavevectors for different values of second- and third-neighbour exchangeconstants for a Heisenberg pyrochlore antiferromagnet with dipolar interactions. Incommensuratestates are indicated by non-zero components of the wavevectors. All transition lines are of thefirst-order.

two Gd–O1–O2–Gd bridges with a significantly larger interoxygen distance |O1–O2| = 3.62 Aand bond angles of 152◦ and 143◦. As a result, the two exchange constants for third-neighbourpairs have to be different with J32 ≪ J31. The Goodenough–Kanamori–Anderson rules alsosuggest that the second-neighbour constant J2 must be smaller than J31 since bond angles inthe corresponding superexchange paths are significantly closer to 90◦. Similar relations shouldhold for next-neighbour exchange constants in Gd2Sn2O7 with, perhaps, a somewhat largerratio J2/J31 due to a larger angle 126◦ in the second-neighbour superexchange path. Theoverall effect of further neighbour exchange is, however, reduced in the stannate because of alarger lattice constant a = 10.45 A compared with a = 10.17 A in the titanate [20].

Next, we consider the following Hamiltonian:

H =!

⟨i, j ⟩Ji j Si · S j + D

!

i

(ni · Si )2 + (gµB)2

!

⟨i, j ⟩

"Si · S j

r3i j

− 3(Si · ri j )(S j · ri j )

r5i j

#,

where the superexchange Ji j extends up to the third-neighbour pairs of spins and D > 0is a single-ion anisotropy. The strength of the dipolar interaction between nearest-neighbourspins Edd = (gµB)2/(a

√2/4)3 is estimated as Edd/J ≈ 0.2 for the titanate [8], where J

is the nearest-neighbour exchange constant (in the following J ≡ 1). Applying mean-fieldtheory [8, 19] and evaluating dipolar sums via Ewald’s summation we have determined theinstability wavevector for different values of second- and third-neighbour exchange constants.Results are essentially independent of the anisotropy constant D > 0 since the dipolarinteraction already selects spins to be orthogonal to the local trigonal axes ni for the eigenstateswith the highest transition temperature.

If only the nearest-neighbour exchange is present, in agreement with previous works [8, 19]we find an approximate degeneracy of modes along the cube diagonal with a very shallowminimum ∼0.5% at k = ( 1

212

12 ). In such a case, a fluctuation-driven first-order transition

is expected to the PC state [9, 21]. The diagram of possible ordering wavevectors for arestricted range of J31 and J2 is presented in figure 4. It contains two commensurate states withk = (0 0 0) and k = ( 1

212

12 ) and three incommensurate phases. Remarkably, already weak

antiferromagnetic J31 robustly stabilizes the k = ( 12

12

12 ) magnetic structure, which exists in a


-0.5 0.0 0.5-0.5

0.0

0.5

(0 0 0)

(q q 0)

(q 0 0)

(q q’ 0)

1 2_ ( _ _ 1 1

2 2 )

J2

J31

Figure 4. Instability wavevectors for different values of second- and third-neighbour exchangeconstants for a Heisenberg pyrochlore antiferromagnet with dipolar interactions. Incommensuratestates are indicated by non-zero components of the wavevectors. All transition lines are of thefirst-order.

two Gd–O1–O2–Gd bridges with a significantly larger interoxygen distance |O1–O2| = 3.62 Aand bond angles of 152◦ and 143◦. As a result, the two exchange constants for third-neighbourpairs have to be different with J32 ≪ J31. The Goodenough–Kanamori–Anderson rules alsosuggest that the second-neighbour constant J2 must be smaller than J31 since bond angles inthe corresponding superexchange paths are significantly closer to 90◦. Similar relations shouldhold for next-neighbour exchange constants in Gd2Sn2O7 with, perhaps, a somewhat largerratio J2/J31 due to a larger angle 126◦ in the second-neighbour superexchange path. Theoverall effect of further neighbour exchange is, however, reduced in the stannate because of alarger lattice constant a = 10.45 A compared with a = 10.17 A in the titanate [20].

Next, we consider the following Hamiltonian:

H =!

⟨i, j ⟩Ji j Si · S j + D

!

i

(ni · Si )2 + (gµB)2

!

⟨i, j ⟩

"Si · S j

r3i j

− 3(Si · ri j )(S j · ri j )

r5i j

#,

where the superexchange Ji j extends up to the third-neighbour pairs of spins and D > 0is a single-ion anisotropy. The strength of the dipolar interaction between nearest-neighbourspins Edd = (gµB)2/(a

√2/4)3 is estimated as Edd/J ≈ 0.2 for the titanate [8], where J

is the nearest-neighbour exchange constant (in the following J ≡ 1). Applying mean-fieldtheory [8, 19] and evaluating dipolar sums via Ewald’s summation we have determined theinstability wavevector for different values of second- and third-neighbour exchange constants.Results are essentially independent of the anisotropy constant D > 0 since the dipolarinteraction already selects spins to be orthogonal to the local trigonal axes ni for the eigenstateswith the highest transition temperature.

If only the nearest-neighbour exchange is present, in agreement with previous works [8, 19]we find an approximate degeneracy of modes along the cube diagonal with a very shallowminimum ∼0.5% at k = ( 1

212

12 ). In such a case, a fluctuation-driven first-order transition

is expected to the PC state [9, 21]. The diagram of possible ordering wavevectors for arestricted range of J31 and J2 is presented in figure 4. It contains two commensurate states withk = (0 0 0) and k = ( 1

212

12 ) and three incommensurate phases. Remarkably, already weak

antiferromagnetic J31 robustly stabilizes the k = ( 12

12

12 ) magnetic structure, which exists in a

A S Wills, et al.. J. Phys.: Condens. Matter 18, L37–L42 (2006).


• k = (1/2 1/2 1/2) • Two non-equivalent

magnetic Gd sites • 3/4 Gd ions ordered

1/4 disordered • possible multi-k

structures • 2nd transition associated

with partial order on disordered site


120x103

100

80

60

40

20

0

-20

Inte

nsity

(co

unts

/ 60

0 s)

1008060402002θ (deg)

1200

800

400

0Inte

nsity

(a.

u.)

1.00.80.60.40.2Temp. (K)

160Gd2Ti2O7

(1/2,1/2,1/2)

(1/2,1/2,1/2)

Figure 1. Rietveld fit at 250 mK to magnetic-only scattering (magnetic R-factor 4%; the lowerline is observed minus calculated intensity). Highlighted on the left of the main figure 1

212

12

magnetic reflection at 420 mK, 540 mK, 670 mK and 750 mK (in order of decreasing intensity).Inset: intensity versus temperature of this reflection, which disappears near the T ′ ≈ 0.7 K phasetransition reported in [7].

Figure 2. Comparison of (left) the 1-k structure and (right) its 4-k variant. In each structure thefour Gd3+ ions coloured orange are shown as carrying no thermally averaged moment. The phasetransition at T ′ = 0.7 K involves weak ordering of these four spins and a small canting of theremaining spins away from the positions shown. At T ≪ 0.7 K, only these ions carry a disorderedspin component.

moments on some Gd3+ ions come out as much greater than 7 µB, which is unphysical. Thisleaves only the 1-k and 4-k structures to be distinguished.

These two structures are compared in figure 2. If the pyrochlore unit cell is describedas four ‘up’ tetrahedra, then the 4-k structure consists of three fully ordered spin tetrahedra(with moment 7.0 µB/spin) and one weakly ordered spin tetrahedron (1.9 µB/spin). Thesederive, respectively, from the ‘planar’ and ‘interstitial’ spin sets of the 1-k structure. In the4-k structure, all spins (in both sets) are perpendicular to the local trigonal ⟨111⟩ axes (thatconnect the vertices of the tetrahedra to their centres), while in the 1-k structure the spins areperpendicular to a single global [111] crystallographic direction. However, the disorderedspin components—which are associated with only the weakly ordered set—have a completelydifferent spatial distribution in the 1-k and 4-k structures. In the 1-k structure, nearest neighbourdisordered spin components are separated by 7.2 Å, while in the 4-k structure they are separated

Lots of diffuse magnetic scattering in the ordered phase

J D M Champion, et. al. Phys. Rev. B 64 140407 (2000)

D20, ILL

COMMENTS PHYSICAL REVIEW B 85, 106401 (2012)

FIG. 1. (Color online) Neutron diffraction data in the low temperature phase of Gd2Ti2O7 at 250 mK. (Top panel) Data with our 4k modelfit published earlier.3 (Middle panel) The same data with the profile expected from the intermediate phase discussed in Ref. 1 (Fig. 12) with themoments forced to be 7 µB .1 (Lower panel) The data and the calculated neutron profile assuming 7µB on the Gd3+ site for the noncollinear,low temperature phase proposed in Ref. 1 (Fig. 13).1 In each panel, the proposed magnetic structure is drawn in relation to the chemical unitcell. In the top panel, the smaller ordered spin is represented by a green ball. Error bars are statistical in nature in the figure and represent ±1σ .

feature of the neutron diffraction data is a considerable amountof diffuse magnetic scattering persisting to low temperature.This is consistent with the incompletely ordered sublattice ofGd moments proposed in our work, but it is inconsistent withthe models proposed in Ref. 1.

The authors of Ref. 1 base their scepticism of our publishedmagnetic structure on two arguments, but in our opinion,neither argument is convincing. First, they assume that theGd3+ ion is in a pure S = 7/2,L = 0 state, and hence is

an ideal Heisenberg spin. However, even in the free ion thispremise is false, as intermediate coupling of angular momentaadmix extra terms into the Gd3+ ground state. In the pyrochlorestructure, the rare-earth (RE) site has trigonal point symmetryand a highly asymmetric local environment of oxide ligands.This induces a local crystal field anisotropy in Gd3+ thatis of a magnitude comparable to the magnetic couplings inthe system. Thus, using ESR, Sosin and coworkers5 found asingle-ion energy gap of 0.25 K. They also report the presence

106401-2

Fully ordered models..

• Efforts to find a fully ordered spinstructure so-far unsuccessful…

Stewart, et. al., PRB 85 (2012) 106401Brammall, et. al., PRB 83 (2011) 054422

AF Pyrochlore - Gd2Ti2O7Cubic symmetry – 4 equivalent k vectors of <½ ½ ½ >

1k structure (as proposed byChampion et. al.)Spins all perpendicular to global<1 1 1> plane

4k structure (only other possibilityconsistent with Gd moment refinement)Spins perpendicular to local <1 1 1> planes connecting vertices oftetrahedra to their centres

J R Stewart, et. al. J. Phys.: Condens. Matter 16 L321 (2004)

7.2 Å3.6 Å


Fitting over 2 NN shells in each model

1k model: poor fit FM correlations (!)

4k model: good fit AFM correlations !The diffuse scattering is able to distinguish between the two models

J R Stewart, et. al. J. Phys.: Condens. Matter 16 L321 (2004)

D7, ILL (pre-upgrade!)

Then we thought we’d check our work…

Isotopically enriched single crystal !Measured on D7, D10, DCS, SPINS and WISH !Found to have 99.4% 160Gd - still gives pretty low transmission !SPINS experiment shows good quality crystal with around 0.4° isotropic mosaic

“You had a 50-50 chance. You weren’t even close.”

Dr Kananga, Live and Let Die

In-field single xtal diffraction

Figure 1: Intensities of magnetic Bragg peaks in Gd2Ti2O7 as a function of applied field B in the low-temperature phase.(a)–(c) are peaks within the (hhl) scattering plane and (d)–(f) are peaks outside the (hhl) scattering plane. The in-planepeaks disappear in small applied field, while the out-of-plane peaks increase in intensity.

a field and the other pair is depopulated. Our data indicate that the field depopulates domain pair (k1

, k4

) which lies inthe scattering plane. The intensity of peaks outside the scattering plane should increase as the other domain pair (k

2

, k3

)is simultaneously populated by the applied field. In practice these out-of-plane peaks proved difficult to locate because ofthe absorbing nature of our sample and the restricted out-of-plane coverage due to the magnet. However, we were able toidentify three out-of-plane magnetic peaks, all of which show an increase of intensity in applied field consistent with the 1kstructure [Fig. 1b]. Our results do not rule out the 2k structure (6 domains along {110} directions, of which one is parallelto the field and perpendicular to the scattering plane) and the 3k structure (4 domains along {111} directions). However,the 4k structure is conclusively ruled out, since it is necessarily single-domain. The only possible scenario in which ourdata are consistent with a 4k structure is if the applied magnetic field actually caused a magnetic phase transition ratherthan a domain imbalance.

Two results suggest that this is unlikely to be the case.First, a magnetic phase transition would be expected tooccur when the Zeeman energy is of the same order of magnitude as the energy scale of magnetic interactions: B ⇠✓CW

kB

/gSµB

⇠ 2T, whereas we observe changes in peak intensity for much smaller fields B ⇠ 0.2T. Second (and moreimportantly), there is no experimental evidence of a phase transition in specific heat measurements for fields along h110iof less than 2T [9].

1.2.2 HRPD experiment

In an attempt to determine conclusively whether the structure is, in fact, 1k, 2k, or 3k, we performed high-resolutionneutron powder diffraction measurement on HRPD (ISIS). The object of this measurement was to look for a distortionif the crystal structure, which must arise from the reduction in symmetry in any non-4k structure. The nature of thisdistortion would distinguish between magnetic structures, since the 1k and 3k structures are rhombohedral whereas the2k structure is orthorhombic. Unfortunately, we were unable to observe any such distortion within the resolution of ourmeasurement. Hence, even though a crystal structure distortion is required by symmetry, its magnitude is too small tomeasure with one of the highest-resolution neutron diffractometers currently available (need lower bound here...). Thisresult reflects the small energy-scale of the magnetic interactions in GTO and the weakness of the magnetoelastic coupling.

3

!

"#!$%&!'()!#**+!,-*!./*01!2(3*#!#(2*!()40*#!5/,-!!!!()1!!

"!6789

!

:!()1!!#!()1!!

$"6;<

!

:+!#%!,-*!1%2(/)#!

)&2=*>!?!()1!8!(>*!*@&/A(0*),! 6()(0%4%&#0$+!)&2=*>!B!()1!C!(>*!*@&/A(0*),! ,%%:D!E*!'())%,!#($!

5-/'-!1%2(/)!F(/>!/#!.(A%&>(=0*!/)!,-*!./*01!6/,!1*F*)1#!%)!()/#%,>%F$!%.!,-*!#&#'*F,/=/0/,$:!=&,!,%!=*!

'%)#/#,*),!5/,-!%&>!*GF*>/2*),+!5*!#-%&01!(##&2*!,-(,!,-*!./*01!F%F&0(,*#!,-*!1%2(/)!F(/>!6?+8:!()1!

1*F%F&0(,*!,-*!6B+C:!F(/>D!

!

"

!

!

!

!

!

Figure 2: Effect of applied magnetic field along [1

¯

10] on the four k domains. The field direction is shown as a red line,the scattering plane is shown in blue, and the four 1k domains are labelled k

1

, k2

, k3

, k4

.

1.3 Aims of this report

The WISH experiment, summarised above, provide very strong evidence that the magnetic structure of GTO is not 4k. Itis therefore incompatible with the conclusion of the previous neutron diffraction paper, which was based on an analysis ofthe magnetic diffuse scattering. The aim of this report is to attempt to resolve this apparent contradiction. To do this, Iwill consider some of the approximations made in Ref. [2], and describe a new analysis of the same data used previously.

My report is set out as follows. I first summarise the argument of Ref. [2] and suggest some possible ways in which itmight the assumptions made might turn out to be invalid. I then describe the reverse Monte Carlo (RMC) method whichI use to test possible structure models against both Bragg and diffuse scattering data. I then consider two “test cases”, inwhich I use simulated data to investigate the information content of the Bragg and diffuse scattering for GTO. Finally,I attempt to fit the experimental data reported in Ref. [2] using the RMC method, and discuss what conclusions can bedrawn from the diffuse scattering.

This isn’t meant as a draft of a paper, but just as a summary of progress. I’ve tried to give as detailed description aspossible of the RMC analysis so far—please let me know what else I can do!

2 Summary of 2004 JPCM

The original analysis was based on neutron powder diffraction data for the low-temperature T < T2

phase. Both theBragg scattering (arising from the average magnetic structure) and magnetic diffuse scattering (arising from correlatedspin fluctuations) were measured.

The argument for the 4k structure proceeded by the following stages:

1. Only the 1k structure and its multi-k variants are realistic candidates to describe the low-temperature physics ofGTO. Other possible magnetic structures can be neglected.

2. The Bragg peak intensities rule out the 2k and 3k variants, since in these structures some spins would have averagemagnetic moments greater than 7.0µ

B

. Hence it remains only to decide between 1k and 4k structures.

3. Although all nk structures have identical powder Bragg diffraction patterns, the diffuse scattering can, in principle,be different between 1k and 4k structures.

4. In the 1k and 4k structures, the diffuse scattering arises entirely from 1/4 of the spins (the interstitial site). Negligiblediffuse scattering arises from the other 3/4 of the spins (kagome site). Hence it’s reasonable to fit the diffuse scatteringincluding only the interstitial sites in the model.

5. The diffuse scattering is, in fact, different enough between the 1k and 4k structures to decide between them.

4

Application of magnetic field along (1-10) results in domain imbalance between!in-plane (k1 and k4) and out-of-plane (k2 and k3) domains!!Hence cannot be 4k structure as k-domains exist

WISH, ISIS

Re-visit diffuse scattering

!"" #$%$#&$ '()*$ +,#-( '(.$--/)0

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T73@ U@ #5:=?6: ;G >;J95< #'+ 1;95??743 G;< '4( 26 S"VH S#V MPP D 249 S$VH S%V BC D@ $R>5<715462? ,S!V S:;?79 ?745VH #'+ 6;62? ,S!V S92:E59?745V J76E 12345678 97GG=:5 :82665<743 S9;6:V 249 ?266785 97GG=:5 :82665<743 S:E;<6 92:E5:V^ 6E5 ?2665< 6J; 2<5 ;GG:56 LI "P@B G;< 8?2<76I@

MnO, T = 15 K

•Co-existance of LRO SRO

•Fluctuations in ordered magnetic state give rise to significant diffuse scattering

Mellergård & McGreevy, Acta Cryst. A55, 783 (1999)

Analysis of disorder in presence of order

All LRO magnetic structures give rise to diffuse scattering due to fluctuations (spin-waves)

Total magnetic moment:

In ordered structures, maximum projection is Sz = ±S with

Therefore we are left with a fluctuating moment of (at least)

So in Gd2Ti2O7 the component of the total moment which is due to fluctuations from the ordered sites is

Ifluct =12S

16S(S + 1)= 0.17

whereas the contribution due to the disordered Gd sites is 1/4.!!So we expect that around 40% of the diffuse scattering intensity is due to the spin-wave scattering

µord

= g2S2µord

= g2S2

µfluct = g2Sµord

= g2S2

µtot

= g2S(S + 1)µord

= g2S2

Reverse Monte Carlo

• “Virtual crystal” • Positions of spins fixed at lattice sites • Orientations of spins refined to fit data

subject to constraints • Average structure constrained to follow

1k model

McGreevy and Pusztai, Mol. Simul. 1, 359 (1988) Paddison & Goodwin, PRL 108, 017204 (2012)

Total scattering calculations

Frey, Acta Cryst. B51, 592 (1995)

I(Q) = IBragg(Q) + Idi↵use(Q)

Si ⌘ hSi+�Si

FluctuationAverageSpin vector

The total scattering can be sub-divided into Bragg and diffuse contributions, where the diffuse scattering now comes from all deviations from perfect order

RMC and partial order

Need to use the D20 data for this, since the resolution and statistics of the D7 data!are not sufficient!!But major advantage of D7 over D20 is unambiguous determination of magnetic !scattering with absolute units!!Therefore, we use the D7 data to scale and align the nuclear Bragg subtracted D20!data to come up with a “poor man’s polarisation analysis”.!!

NB: desperately need PA on good resolution diffractometers with wide Q-range

1.4 K D7 data

D20 data

RMC and partial orderRMC is now performed, with the Bragg intensity constrained to be either 1k or 4k

1k structure 4k structure

D20, ILL

Correlations from RMC

Ordered-Ordered Disordered-Disordered


120x103

100

80

60

40

20

0

-20

Inte

nsity

(co

unts

/ 60

0 s)

1008060402002θ (deg)

1200

800

400

0Inte

nsity

(a.

u.)

1.00.80.60.40.2Temp. (K)

160Gd2Ti2O7

(1/2,1/2,1/2)

(1/2,1/2,1/2)


212

12






120x103

100

80

60

40

20

0

-20

Inte

nsity

(co

unts

/ 60

0 s)

1008060402002θ (deg)

1200

800

400

0Inte

nsity

(a.

u.)

1.00.80.60.40.2Temp. (K)

160Gd2Ti2O7

(1/2,1/2,1/2)

(1/2,1/2,1/2)


212

12





1k

4k

r1

r1

Ordered-disordered

r1

Correlations from RMC

Predictions for multi-k• Latest work predicts 1k / 4k

structure between TN1 and TN

2

• Possible 2k state at low T

4

to 1-k states including (±1, 0, 0, 0), (0, ±1, 0, 0), etc. Simi-larly, there are sixteen “ 4-k points” on the hyper-sphere [53].Given a certain spin configuration, we calculate ˆ ⌘ /| |.The d

ik (i = 1, 4) are then defined as the minimum Euclideandistance between point ˆ and all of the i-k points. The ther-mal average of d1k (d4k) is expected to decrease if the systementers a 1-k (4-k) state for T < T

c

.The results for two sets of interaction parameters in the

1-k and 4-k regions of Fig. 2 are shown in the left andright columns of Fig. 3, respectively. The growth of | |2at T

c

/J ⇠ 5.5 and Tc

/J ⇠ 0.15 shows that the system orderswith k = (

12

12

12 ). In the left column, the 1-k state is selected

at Tc

, as indicated by a minimum for d1k and a maximum ford4k. The system orders in a 4-k state in the right column. Theseparation of d1k and d4k for both cases in Fig. 3 accentuatesas the linear dimension L of the system increases, indicatingthat the selection of either 1-k or 4-k survives in the thermo-dynamical limit. These results are consistent with the predic-tions from the E-TAP calculations at T

c

. Unfortunately, thelarge computational resources required for simulations withlong-range dipolar interactions prevent us from investigatingthe order of the phase transitions in Fig. 3.

The kinks in | |2 and merging of d1k and d4k indicate thesystem enters into a distinct phase at T/J <⇠ 4 and T/J <⇠ 0.1in the left and right columns of Fig. 3. Since in a 2-k stated1k ⌘ d4k =

p2 � p

2 ' 0.765 [53], the results for dik sug-

gest that the low-temperature region may be a 2-k state. Wenote that the latest single-crystal neutron diffraction results[63] indicate that the low-temperature state (T < 0.7 K) ofGd2Ti2O7 may not be the previously suggested 4-k structure[20]. The results of a more in-depth numerical investigationof the low-temperature regime will be reported elsewhere.

Discussion – Considering a general symmetry-allowedanisotropic Hamiltonian, we found that k = (

12

12

12 ) partial or-

der can occur over a wide range of anisotropic exchange andlong-range dipolar interactions in pyrochlore magnets. We ar-gued that fluctuations beyond s-MFT are responsible for thestabilization of a 1-k or 4-k partially ordered structure. Thisconclusion is based on results from E-TAP calculations whereon-site fluctuations are included. We used Monte Carlo sim-ulations to illustrate that different values of the magnetic ex-change interactions can, as anticipated on the basis of the E-TAP calculations, lead to either 1-k or 4-k order.

From our work, we have exposed a likely mechanism forthe establishment of 4-k order in Gd2Ti2O7 below its para-magnetic transition [20]. Further quantitative progress on thisproblem will require a better estimate of the material exchangeparameters [48, 52, 64]. From this work we conclude that thetransition from the paramagnetic state to the 4-k phase shouldbe second order and belong to the 3D n = 4 cubic universalityclass. Experimental evidence [33] suggests that this transitionis second order in Gd2Ti2O7. Determining the critical expo-nents for this system could confirm our prediction but will bea challenge, given that the exponents for the 3D n = 4 cu-bic and 3D Ising, XY and Heisenberg universality classes areproximate to one another [47, 58].

0.1 0.15 0.2 0.25

0.2

0.4

0.6

0.1 0.15 0.2 0.250.2

0.4

0.6

0.8

T/J

4 6 8

0

0.2

0.4

0.6

4 6 80.2

0.4

0.6

0.8

T/J

d1k L−4d1k L−6d1k L−8d4k L−4d4k L−6d4k L−8

d1k L−4d1k L−6d1k L−8d4k L−4d4k L−6d4k L−8

L−4L−6L−8

L−4L−6L−8| |2

dik

1

FIG. 3. (Color online) Monte Carlo simulations results. Top row:growth of | |2. Bottom row: value of dik as a function of T/J . Leftcolumn: Jpd = 0, Jdip/J = 0.18, J2/J = �1, JIsing/J = 10,JDM/J = 10 (direct DM) and J3/J = 1 (a point in 1-k region).Right column: Jpd = JDM = JIsing = 0, Jdip/J = 0.18 andJ2/J = �0.02 (a point in 4-k region). Note: The dik are nonzeroin the paramagnetic phase because their value is equal to the averagedistance of a random point on a 4-dimensional hypersphere from the1-k and 4-k points.

Finally, the E-TAP method for frustrated magnets formu-lated here could prove useful for other systems where stateselection at T

c

is an open question. This is particularly soif the low-temperature state selected via (thermal or quan-tum) ObD is separated by a phase transition from the stateselected at T

c

. In such a case, an understanding of ObD atT = 0

+ can not be leveraged to explain the selection at Tc

.Examples include the transition to long-range order in the py-rochlore Heisenberg antiferromagnet with indirect DM inter-actions [50, 65, 66], the problem of magnetization directionselection in face-centered cubic dipolar ferromagnets [67, 68]and the topical issue of state selection in XY pyrochlore anti-ferromagnets [45, 49, 64, 69–73].

We thank Steve Bramwell, Alexandre Day, Jason Gardner,Paul McClarty, Oleg Petrenko, Rajiv Singh, Ross Stewart, Et-tore Vicari and Andrew Wills for useful discussions. We ac-knowledge Pawel Stasiak for his help with the Monte Carlosimulations and Peter Holdsworth for his comments on themanuscript. This work is supported by the NSERC of Canada,the Canada Research Chair program (M.G., Tier 1) and by thePerimeter Institute for Theoretical Physics. Research at PIis supported by the Government of Canada through IndustryCanada and by the Province of Ontario through the Ministryof Economic Development & Innovation.

3

Eq. (2) is the celebrated n-vector model (n = 4) with cubicanisotropy [46] with r = a0(T �T

c

) and a0 > 0. This model,a cornerstone of the theory of critical phenomena [46, 47], hasbeen extensively investigated with numerous methods [55–58]. In 3D, for n > n

c

⇡ 2.89 and u > 0 [58], the modelundergoes a second-order transition into a phase where all

a

have the same amplitude if v > 0. The universality classis controlled by the cubic fixed point [46] with distinct crit-ical exponents from those of the isotropic O(n) fixed point[47, 58]. For v < 0, the phase transition is a fluctuation-induced first-order transition to a state with only one nonzero a

. Therefore, we conclude that for v > 0 the magnetic orderis defined by a superposition of four k

a

spin states (4-k state)while for v < 0 the order is defined by a single k

a

spin struc-ture (1-k state). Both states are POSs since the spins on 1/4

of the sites remain disordered (see Fig. 1). As stated above, s-MFT predicts that 1-k and 4-k have the same F (i.e. v = 0).We thus identify thermal fluctuations as the mechanism forgenerating v 6= 0 leading to a selection of 1-k vs 4-k. Toexpose how fluctuations may lead at the microscopic level toa selection directly at T

c

, we devise and then use an E-TAPmethod [59–61].

Extended TAP Method – A magnetic moment at a particularlattice site experiences a local field due to its neighbors. At thes-MFT level, the presence of the spin at the site of interest af-fects its local field indirectly. This is an artifact of s-MFT. TheOnsager reaction field (ORF) introduces a term that cancelsthis unphysical effect. The TAP approach provides a system-atic way to implement the ORF correction [62]. In this work,we devise an extension of the TAP method, E-TAP, where theORF is the first term of a series originating from nonzero on-site fluctuations, e.g. hS↵

i

S�

i

i 6= hS↵

i

ihS�

i

i [53, 60, 61].To compute the E-TAP corrections, we consider a perturba-

tive expansion of the Gibbs free-energy, G, in inverse temper-ature � [53, 60].

�G = � ln

⇣Tr[exp

� � �H +

X

i

�i

· (Si

� mi

)

�]

⌘. (3)

Here mi

is the local magnetization and the �i

vector is aLagrange multiplier; different from the local mean field atsite i by a factor of � [53, 60]. Defining ˜G(�) ⌘ �G, thefirst and second terms in the expansion, ˜G(0)/� and ˜G0

(0),where the prime represents differentiation with respect to �,are the s-MFT entropy and energy, respectively. The thirdterm, ⌦ ⌘ ˜G00

(0)�/2, is the first correction beyond s-MFT,arising from fluctuations [53, 60]:

⌦ = ��

4

X

i,j

X

↵��

J↵�

ij

J��

ij

�↵�

i

��

j

. (4)

Here, �↵�

i

= hS↵

i

S�

i

i � hS↵

i

ihS�

i

i is the on-site suscepti-bility. Since �↵�

i

is quadratic in { a

} [53], ⌦ in Eq. (54)is therefore of quartic order in {

a

}. The first order E-TAPcorrection, ⌦, can thus, in principle, generate a finite cubicanisotropy term in F and select 1-k or 4-k depending on

��

�

�

�

�

�

JIsing

J

k = 0

k = 0

4-k�0.15

0.15

1-k

JDM

J

1

FIG. 2. (Color online) Ordering wave vectors at T = Tc ob-tained from s-MFT. The combined area denoted 1-k and 4-k displays( 12

12

12 ) order but 1-k and 4-k are degenerate at the s-MFT level. The

first order E-TAP correction Eq. (54) applied in this regime selectseither 1-k or 4-k. The k = 0 region encompass all states for whichall sites are fully ordered and each primitive 4-site tetrahedron basishas the same local spin order. The dashed (0.15) and dotted (-0.15)contours mark the boundaries for the corresponding Jpd values.

the bilinear spin-spin interaction matrix J↵�

ij

defined throughH ⌘ P

(i>j);↵,� J↵�

ij

S↵

i

S�

j

in Eq. (1).We calculate ⌦ [53] for the region in parameter space with

(

12

12

12 ) ordering (the dark-shaded 1-k/4-k wedge in Fig. 2).

Specifically, we compute �⌦ ⌘ ⌦1k � ⌦4k. �⌦ < 0 indi-cates a 1-k selection and conversely for �⌦ > 0, with the⌦ contribution splitting the wedge into 1-k and 4-k sectors.In particular, for a dipolar Heisenberg model with JIsing =

JDM = Jpd = 0, relevant to Gd2Ti2O7 [21], E-TAP calcu-lations predict a 4-k state selection at T

c

, as observed in thiscompound [20].

Monte Carlo Simulations – We performed parallel temper-ing classical Monte Carlo simulations to check the E-TAP pre-dictions. We pick, somewhat arbitrarily, two sets of interac-tion parameters corresponding to the 1-k and 4-k regions inFig. 2. One set is the simplest model for a spin-only 8S7/2

(L = 0, S = 7/2, J = L + S) state for Gd3+ in Gd2Ti2O7,with Jdip/J = 0.18 [21], JIsing = JDM = Jpd = 0. Weinclude, as in a previous Monte Carlo work [25], weak ferro-magnetic second n.n. interaction (e.g. �0.02J <⇠ J2 < 0)to stabilize a temperature range wide enough to numericallyresolve (

12

12

12 ) order [25, 53]. We measure the magnitude of

:

| |2 =

*3X

a=0

2a

+, (5)

where h. . .i denotes a thermal average.In addition to its magnitude, we identify the orientation of

at T < Tc

through two additional order parameters, dik

(i = 1, 4) defined below. On the surface of a four-dimensionalunit hyper-sphere, there are eight “1-k points” corresponding

Javanparast, et. al., arxiv:1310.5146

Structure of GTOSo in the end powder data (inc. diffuse scattering) completely indecisive in selection of multi-k magnetic structure

If 1k structure is right (is it now appears) then magnetic transition involves a loss of symmetry from cubic - rhombohedral. Must be an accompanying structural distortion - have looked for this on HRPD. Hope to continue the search on TRISP.

Message: must include all sources of fluctuations in analysis of partially ordered magnetic structures, which are common in frustrated systems

RMC is an ideal way to do this

Low temperature structure is complex - and involves both Gd sites

Moral of the story…“The best scientific theories are those which can be proved wrong”

Karl Popper Thanks for listening!

“The best scientists are those who prove themselves wrong”

“the (low t) magnetic ground state of gd2ti2o7 is not 4k” · alex hannon arwel hughes ann terry...

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