REPORTS
V2487 Oph and 1RXS J173200.0-1934 arecloser than the angular resolution of the
XMM-Newton instruments (i.e., they are lessthan ⬃6 arc sec apart). The ROSAT spectrum(exposure for 462.7 s, with counts in theenergy range between 0.75 and 2.4 keV ) is
compatible with the XMM-Newton spectrum. The 1RXS J173200.0-1934 flux, 3.3 ⫻10⫺13 to 8.9 ⫻ 10⫺13 erg cm⫺2 s⫺1, from theRASS bright source catalog (1RXS-B) is
consistent with that from XMM-Newton(9.4 ⫻ 10⫺13 erg cm⫺2 s⫺1). The similar
fluxes indicate that 1RXS J173200.0-1934
and V2487 Oph might be the same object.
Therefore, the site of a classical nova may
have been seen before the nova explosion
itself (as early as 8 years before it), thanks to
the x-rays emitted in the accretion flow. This
supports the cataclysmic variable scenario fornova explosions.
References and Notes

1. S. Nakano, K. Takamizawa, R. Kushida, Y. Kushida, T.
Kato, Int. Astron. Union Circ. 6941 (1998).
2. A. V. Filippenko et al., Int. Astron. Union Circ. 6943
(1998).
3. W. Liller, A. Jones, Inform. Bull. Var. Stars 4870, 1(2000).
4. J. Krautter, H. Ögelman, S. Starrﬁeld, R. Wichmann, E.
Pfeffermann, Astrophys. J. 456, 788 (1996).
5. H. Ögelman, M. Orio, J. Krautter, S. Starrﬁeld, Nature
361, 331 (1993).
6. L. Shanley, H. Ögelman, J. Gallagher, M. Orio, J.
Krautter, Astrophys. J. 438, L95 (1995).
7. M. Orio, J. Greiner, Astron. Astrophys. 344, L13(1999).
8. The nuclear burning time scale is deﬁned as the timeneeded to burn by nuclear reactions (of hydrogen, in
novae) all of the remaining envelope. Its value isabout 400 years for an envelope mass of 10⫺4 MJ
and a luminosity of 2 ⫻ 104 LJ, and it is proportional
to mass and inversely proportional to luminosity.
9. R. D. Gehrz, J. W. Truran, R. E. Williams, S. Starrﬁeld,
Pub. Astron. Soc. Pac. 110, 3 (1998).
10. M. Orio, J. Covington, H. Ögelman, Astron. Astrophys.
373, 542 (2001).
11. S. N. Shore, S. Starrﬁeld, G. Sonneborn, Astrophys. J.
463, L21 (1996).
12. R. González-Riestra, M. Orio, J. Gallagher, Astron.
Astrophys. 129, 23 (1998).
13. K. M. Vanlandingham, G. Schwarz, S. N. Shore, S.
Starrﬁeld, Astron. J. 121, 1126 (2001).
14. NASA High Energy Astrophysics Science Archive Research Center (HEASARC) (http://xmm.gsfc.nasa.gov/
cgi-bin/Tools/w3nh/w3nh.pl).
15. Parameters of the model best ﬁtting the spectra of
V2487 Oph follow, with 1␴ uncertainties. Low-T
MEKAL: kT (keV) ⫽ 0.22 ⫾ 0.01, emission measure2
2
55 cm⫺3; high-T(3.1–⫹0.3
0.2 ) ⫻ 10 (d/10 kpc) ⫻ 10
MEKAL: kT (keV) ⱖ 48, emission measure 3.5(⫾0.9) ⫻ 102 (d/10 kpc)2 ⫻ 1055 cm⫺3; Gaussianline: Eline (keV) ⫽ 6.5 ⫾ 0.1, ␴line (keV) ⫽ 0.6 ⫾ 0.1,⫹1.3) ⫻ 10⫺5. BlackFline ( photons cm⫺2 s⫺1) ⫽ (6.6–1.1
body temperature and absorption NH have been ﬁxed
(30 eV and 4 ⫻ 1021 cm⫺2, respectively). The emission measure of the MEKAL plasma model is deﬁned
as 兰nenHdV (where ne and nH are the electron andhydrogen densities, in cm⫺3, and V is the volume ofthe emitting plasma). The bolometric ﬂux of theblackbody component is 2.9 (⫾0.7) ⫻ (d/10 kpc)2 ⫻1037 erg s⫺1, and the total absorbed ﬂux, in the 0.3
to 8.0 keV range, is 4.6 ⫻ 10⫺12 erg cm⫺2 s⫺1,yielding an x-ray luminosity of 5.2 ⫻ (d/10 kpc)2 ⫻1034 erg s⫺1. The reduced ␹2 of the ﬁt is ␹2␯ ⫽ 1.18,
with number of degrees of freedom ␯ ⫽ 340.
16. The distance to V2487 Oph derived by Lynch et al.
(30) was 27 ⫾ 3 kpc. They used an empirical MVmax(t2 )relation to obtain MVmax (absolute visual magnitude at

maximum), the apparent visual magnitude at maximum mVmax from observations (1), and the visualextinction, AV ⫽ 1.16 ⫾ 0.24 mag; they obtained AV
from the color excess E(B-V) ⫽ 0.38 ⫾ 0.08 obtained
from the observed OI ␭8446 and ␭11287 near-infrared lines. The distance 27 kpc is very large, and its
determination should be taken with caution. It is
probable that for very fast novae (such as V2487
Oph), the real visual maximum is missed by observers
and therefore the true mVmax is much brighter. The
faintness of the nova (mV ⫽ 9.5 at discovery) further
prevents its discovery at maximum. An estimation ofmVmax follows from extrapolation of the visual light
curve back in time to the prediscovery data or tosome reasonable mVmax, with a slope dictated by t2 or
t3 (31). This exercise yields mVmax in the range 6.8 to
9.5, leading to a distance between 8 and 27 kpc (forMVmax ⫽ – 8.8 and AV ⫽ 1.16). Another source of
uncertainty is the empirical MVmax(t2 ) relation, whichshould not necessarily be valid for all the systems(although it is generally accepted as such).
17. J. MacDonald, S. Vennes, Astrophys. J. 373, L51(1991).
18. S. Balman, J. Krautter, H. Ögelman, Astrophys. J. 499,
395 (1998).
19. S. Balman, J. Krautter, Mon. Not. R. Astron. Soc. 326,
1441 (2001).
20. The nova GQ Mus has the longest turnoff timeobserved to date, 9 years (5), and nova LMC 1995
was still emitting soft x-rays in 2000 (32). These twonovae and V2487 Oph are the only ones to date withobserved (through x-rays) turnoff times longer than 3
years.
21. R. Mewe, J. S. Kaastra, D. A. Liedhal, Legacy 6, 16(1995); available at http://heasarc.gsfc.nasa.gov/
docs/journal/meka6.html.

22. K. Arnaud, B. Dorman, XSPEC User’s Guide for Version
11.1.x (HEASARC, NASA/GSFC, Greenbelt, MD, 2001).
23. J. Patterson, J. C. Raymond, Astrophys. J. 292, 535(1985).
24. M. Ishida, K. Mukai, J. Osborne, Publ. Astron. Soc. Jpn.
46, L81 (1994).
25. C. Hellier, K. Mukai, M. Ishida, R. Fujimoto, Mon. Not.
R. Astron. Soc. 280, 877 (1996).
26. R. Fujimoto, M. Ishida, Astrophys. J. 474, 774 (1997).
27. I. Hachisu, M. Kato, T. Kato, K. Matsumoto, preprint
available at http://xxx.lanl.gov/astro-ph/0110265.
28. M. Livio, A. Shankar, J. W. Truran, Astrophys. J. 330,
264 (1988).
29. W. Voges et al., Astron. Astrophys. 349, 389 (1999);
available at www.xray.mpe.mpg.de/rosat/survey/
rass-bsc.
30. D. K. Lynch, R. J. Rudy, S. Mazuk, R. C. Puetter,
Astrophys. J. 541, 791 (2000).
31. M. J. Harris et al., Astrophys. J. 522, 424 (1999).
32. M. Orio, J. Greiner, W. Hartmann, M. Still, in ClassicalNova Explosions, M. Hernanz and J. José, Eds., AIP
Conference Proceedings, vol. 637 (AIP, New York,
2002), pp. 355–359.
33. S. Snowden, M. Still, I. Harrus, M. Arida, P. Brendan, An
Introduction to XMM-Newton Data Analysis (NASA/
GSFC XMM-Newton Guest Observer Facility, Greenbelt, MD, 2001).
34. This work is based on observations obtained with
XMM-Newton, a European Space Agency (ESA) science mission with instruments and contributions directly funded by the ESA member states and by
NASA. Supported in part by the Ministerio de Ciencia
y Tecnologı́a.
17 July 2002; accepted 6 September 2002

Magnetic Superstructure in the
Two-Dimensional Quantum
Antiferromagnet SrCu2(BO3)2
K. Kodama,1 M. Takigawa,1* M. Horvatić,2 C. Berthier,2,3
H. Kageyama,1 Y. Ueda,1 S. Miyahara,1,4 F. Becca,4 F. Mila4
We report the observation of magnetic superstructure in a magnetization plateau
state of SrCu2(BO3)2, a frustrated quasi–two-dimensional quantum spin system.
The Cu and B nuclear magnetic resonance (NMR) spectra at 35 millikelvin indicate
an apparently discontinuous phase transition from uniform magnetization to amodulated superstructure near 27 tesla, above which a magnetization plateau
at 1/8 of the full saturation has been observed. Comparison of the Cu NMR
spectrum and the theoretical analysis of a Heisenberg spin model demonstrates
the crystallization of itinerant triplets in the plateau phase within a large
rhomboid unit cell (16 spins per layer) showing oscillations of the spin polarization. Thus, we are now in possession of an interesting model system to study
a localization transition of strongly interacting quantum particles.
The competition between itinerancy favoredby kinetic energy and localization favored byrepulsive interactions is a fundamental aspect
1
Institute for Solid State Physics, University of Tokyo,
Kashiwa, Chiba 277-8581, Japan. 2Grenoble High
Magnetic Field Laboratory, CNRS and MPI-FKF, 38042
Grenoble, France. 3Laboratoire de Spectrométrie Physique, Université Joseph Fourier Grenoble I, 38402
St.-Martin d’Hères, France. 4Institut de PhysiqueThéorique, Université de Lausanne, CH-1015 Lausanne, Switzerland.

*To whom correspondence should be addressed. Email: masashi@issp.u-tokyo.ac.jp

of many-body quantum systems, of which theMott (metal-insulator) transition is an example (1). Similar phenomena may occur for the
spin degrees of freedom of certain magnetic
insulators, known as spin liquids, in which
the ground state is a singlet separated from
triplet excitations by a finite energy gap ⌬.
Transition metal oxides exhibiting this kindof physics have been actively studied in aneffort to understand the more complex physics of high-temperature superconductors[e.g., (2)], a good example being spin ladders(3). In such systems, a magnetic field Hc ⫽www.sciencemag.org SCIENCE VOL 298 11 OCTOBER 2002

395

REPORTS⌬/g␮B (where g is the effective g-value of theelectron spin and ␮B is the Bohr magneton
will close the gap, and the resulting magnetization is a gas of mobile triplets whose
density can be tuned by the field value. However, when this density becomes commensurate with the underlying crystal lattice, thetriplets may crystallize into a superlattice.
The magnetization density will then stay constant for a certain range of magnetic field,
resulting in a magnetization plateau (4, 5),which has indeed been observed in somequantum spin systems (6, 7). It was argued
theoretically that a magnetization plateau at afractional value of 1 ␮B per unit cell occurs
only when translational symmetry of thecrystal lattice is broken by a magnetic superlattice (8). We report direct observation ofFig. 1. (A) The schematic structure of
SrCu2(BO3)2 viewed
along the c axis. The
box shows a unit cellprojected on the abplane. The structure
consists of a stack ofalternating
Cu-B-O
magnetic layers andnonmagnetic Sr layers. See (9) for details.
(B) The Shastry-Sutherland spin model with the intradimer exchange J and the interdimer exchange J⬘, which is
topologically equivalent to the
Cu network within a layer of
SrCu2(BO3)2. (C and D) Proposed
conﬁgurations of triplets for the
1/8 magnetization plateau phase(21). Solid circles denote thetriplet dimers; numbers distinguish inequivalent Cu sites.

such a magnetic superlattice by NMR experiments in SrCu2(BO3)2, a quasi–two-dimensional quantum spin system.
The planar network of orthogonal dimersof spin 1/2 Cu2⫹ ions in SrCu2(BO3)2 (Fig.
1A) (9) has stimulated extensive experimental (10 –16) and theoretical (17–23) work.
This compound is a realization of the
Shastry-Sutherland model (Fig. 1B) conceived theoretically two decades ago (24):H⫽J

冘

共n.n.)

S i 䡠 S j ⫹ J⬘冘Si 䡠 Sj(1)(n.n.n.)where J and J⬘ are respectively the nearest
neighbor (intradimer) and the next nearest
neighbor (interdimer) antiferromagnetic exchange constants between two S ⫽ 1/2 spins
Si and Sj. When J is much larger than J⬘, themodel reduces to a collection of dimers. In
the opposite limit, J/J⬘ ⬍⬍ 1, it is equivalent
to the antiferromagnetic Heisenberg model
on a square lattice with a Néel orderedground state. The Shastry-Sutherland modelhas the property that the direct product of the
dimer singlet states is the exact ground state
when J⬘/J is smaller than a certain criticalvalue near 0.7 (17, 18, 22, 24). An excitedstate can be made by promoting a singlet
dimer into a triplet state, which can then hopfrom one dimer to another when J⬘ ⫽ 0. The
frustration between J and J⬘ in the ShastrySutherland model, however, inhibits suchhopping up to fifth order of the perturbation
series in J⬘/J, leading to extremely flat triplet
bands (i.e., a very small kinetic energy) (18).
Various experiments have established thatthe dimer singlet ground state (10, 12) is
indeed realized in SrCu2(BO3)2, with an energy gap to magnetic excitations of 3.0 meV(13, 14). Nearly flat triplet bands were also
observed by neutron inelastic scattering experiments (14, 15). The most striking property of SrCu2(BO3)2 is the plateaus in themagnetization curve at 1/8, 1/4, and 1/3 of the
fully saturated moment (10, 11). One can
expect that the small kinetic energy of tripletsin the Shastry-Sutherland model will allow
them to crystallize into a superlattice at thesecommensurate densities if there are repulsive
interactions between triplets. This is analogous to the Wigner crystallization or chargeordering of electron systems. Although theories have indeed supported such a picture(19 –21, 23) and predicted possible structuresof the superlattices (19 –21), no experimental
evidence has yet been reported. InequivalentCu sites in a commensurate magnetic superstructure should have different hyperfine

Fig. 2. Cu NMR spectra (blackdots) at 35 mK obtained byusing a 20-MW resistive magnet at the Grenoble High
Magnetic Field Laboratory
with a dilution refrigerator.
The integrated intensity of Cu
spin-echo signal is plotted at
discrete frequencies. The inset
shows the spectrum at H0 ⫽26 T. The line is a ﬁt to Eq. 2(see text). The arrows indicate
the peak positions of the ﬁtting. The main panel shows
the spectrum at H0 ⫽ 27.6 T.
The red line is a ﬁt by the sumof contribution from 11 distinct sites, each of which isrepresented by Eq. 2. The parameter values are listed intable S1. The arrows indicate
the resonance frequencies for
Hn ⫽ 0. The strong signal in
the range 371 to 379 MHz
comes from 11B nuclei; thepeak at 125.6 MHz is due to 10B nuclei. The sensitivity of the NMR spectrometer was calibrated using 11B signal at 2 K at several frequencies. Because ofunavoidable local variation of the sensitivity, the accuracy of the intensity (vertical) scale is estimated to be typically ⫾20%.

396

11 OCTOBER 2002 VOL 298 SCIENCE www.sciencemag.org

REPORTS
fields that manifest as distinct NMR lines.
Therefore, we performed NMR measurements on a single crystal of SrCu2(BO3)2
grown by the traveling solvent–floating zone
method (25) in a magnetic field up to 28 T
applied along the c axis at 35 mK. This
covers the first “1/8” magnetization plateau,
which has been observed (11) in the field
range 27 to 28.5 T.
We show in Fig. 2 the Cu NMR spectra ina field of 26 T (inset) and 27.6 T (mainpanel). A single Cu site gives six NMR lines
at the following frequencies (26),f (␣,m) ⫽ ␣␥[(1⫹K) H 0⫹H n] ⫹ ␣␯c(m⫺1/2)(2)
The first term is the Zeeman frequency due tothe external field H0 corrected for the orbital
shift K ⫽ 1.69% (12) and the hyperfine fieldHn; ␣␥ (␣ ⫽ 63 or 65) is the known gyromagnetic ratio of either 63Cu or 65Cu nuclei.
In general, the spin polarizations, both on thesame site and on the neighboring sites contribute to Hn. Here we assume only the dominant on-site coupling, Hn ⫽ Acgc具Sz典, where
具Sz典 is the time-averaged local magnetization.
The coupling constant and the g value werepreviously determined as Ac ⫽ –23.8 T/␮B
and gc ⫽ 2.28 (12, 13). The second term
represents the quadrupolar shift for spin 3/2
nuclei, where m ⫽ 3/2, 1/2, or –1/2 distinguishes three transitions (Iz ⫽ m 3 m ⫺ 1)
and 63␯c ⫽ 22.1 MHz, 65␯c ⫽ 20.5 MHz as
previously determined (12). The Cu NMR
spectrum at 26 T can be well fitted by Eq. 2
with a Gaussian distribution of Hn peaked at–1.79 T with a full width at half maximum(FWHM) of 0.59 T, demonstrating that thesystem at 26 T is in the uniform phase (具Sz典 ⫽0.033) where triplets are itinerant.
At a field of 27.6 T, a drastic change ofthe spectrum was observed with the appearance of many sharp peaks distributed over awide frequency range, indicative of a commensurate magnetic order with a large unit
cell. This is the first clear evidence for amagnetic superlattice in the 1/8 plateau phasewith broken translational symmetry. Although the spectrum appears complicated,some important features are immediately recognized. The sharp six peaks at low frequencies, shown by the red zone in Fig. 2, have theintensity approximately 1/8 of the entire
spectrum and their positions are given by Eq.
2 with 具Sz典 ⫽ 0.30. Likewise, the broad lines
in the yellow zone in Fig. 2 can be ascribed toa single site with 1/8 population and 具Sz典 ⫽0.20. If the triplets were confined on a single
dimer, 1/8 of the Cu sites would have 具Sz典 ⫽0.50 and 具Sz典 ⫽ 0 for the rest. Therefore, themagnetization must be spread over several
sites within the magnetic unit cell. We tried to
fit the entire spectrum by progressively adding a new set of lines given by Eq. 2 with a
Lorentzian distribution of Hn until all the
peaks and the overall shape were reproduced.
We found that at least 11 distinct Cu sites
had to be included. The population of eachsite can be chosen to be an integral multipleof 1/16, which is compatible with a magnetic unit cell containing 16 spins. A successful fit is shown by the red line in Fig. 2,

Fig. 3. (A) Magnetization proﬁle obtained as described in the text. Red circles indicate positive
具Sz典, blue circles indicate negative 具Sz 典, and circle size represents the magnitude of 具Sz 典. The
calculated values for 具Sz 典 are 0.386 (site 1), 0.010 (site 2), – 0.0063 (site 3), – 0.215 (site 4),
0.0073 (site 5), 0.312 (site 6), 0.040 (site 7), and – 0.035 (site 8). (B) Histogram of thehyperﬁne ﬁeld 具Hn典. The middle panel (red) shows the results of the ﬁtting of the NMR
spectrum (Fig. 2, red line) (table S1). Long lines indicate that the population of the site is 1/8;
short lines indicate that the population of the site is 1/16. The top panel is obtained from thetheoretical magnetization proﬁle in (A) assuming only the on-site hyperﬁne coupling A ⫽ Ac ⫽–23.8 T/␮B as described in the text. Generally there are also transferred hyperﬁne couplings toneighboring spins, denoted B and C in (A), but these are much smaller than the on-site coupling
and are difﬁcult to estimate. The bottom panel is obtained from the theoretical proﬁle in (A)assuming B ⫽ –1.04, C ⫽ –2.45 T/␮B. [The previous low-ﬁeld data (12) put the constraint A ⫽Ac – 4B – C.] The improved agreement with experiments, however, is only indicative.

and the corresponding parameters are listed
in table S1. The histogram of the peakvalues of Hn is shown in the middle panelof Fig. 3B. Several sites have positive Hn(blue zone in Fig. 2), implying (through anegative hyperfine coupling) that spins arepolarized opposite to the magnetic field
direction. Thus, the magnetization oscillates within the unit cell. We confirmed thatthe average of Hn over all the sites give thespatial average of 具Sz典 to be 0.063, exactly
1/8 of the saturation.
Let us examine whether our results arecompatible with the Shastry-Sutherlandmodel. In the simplest theoretical approach,interactions between two triplet dimers arecalculated by a perturbation in J⬘/J (19 –21). Two different configurations (Fig. 1, C
and D) have been proposed for the 1/8
plateau (21). Here, each triplet is completely localized on one dimer, the result of an
oversimplification of this approach. The
real magnetization of the Shastry-Sutherland model spreads over all the sites, as
explained below. The number of inequivalent Cu sites, however, does not depend onsuch details of the magnetization profileand can be determined only from symmetry
considerations. For the square cell, there
are six distinct Cu sites, as shown in Fig.
1C. The number of Cu sites may increasedepending on how the layers stack along
the c axis, however, only up to eight, which
is clearly not sufficient to reproduce all theobserved peaks. On the other hand, therhomboid cell has eight Cu sites for a single
layer (Fig. 1D) and the stacking of layers
may increase the number up to 16, although
all the sites may not be resolved experimentally. Therefore, we may expect that
only the rhomboid cell is compatible withthe experiments.
To go beyond these qualitative considerations and check whether the Shastry-Sutherland model can indeed reproduce the peculiar
spin texture revealed by NMR, with positive andnegative magnetizations, we must solve theoriginal spin Hamiltonian of Eq. 1 for a finitesize lattice. There is a subtlety, though. If theground state breaks the translational symmetry,it is expected to be eightfold degenerate for an
infinite system and the ground-state magnetization is not well defined. In the actual compound,a unique ground state would be selected bypinning from impurities or from a lattice distortion. In our finite-size calculation, although periodic boundary condition partially lifts theeightfold degeneracy, some mechanism stillmust be added to the model to enable selectionof one of the ground states (27). Because thepronounced softening of the sound velocity observed at the edges of the magnetization plateaus (16) suggests that a lattice distortion indeed occurs in SrCu2(BO3)2, we decided toinclude an adiabatic spin-phonon coupling (28),www.sciencemag.org SCIENCE VOL 298 11 OCTOBER 2002

397

H⫽

冘冉
冘冉

J 1⫺␣

共n.n.兲

K⫹
2

共n.n.兲

冊冊

␦d ij
S i 䡠 S j⫹d ij0

储␦r i ⫺ ␦r j 储
d ij0

冘冉
冘冉

J ⬘ 1⫺␣⬘

共n.n.n.兲

K⬘⫹
2

共n.n.n.兲

REPORTS

冊
冊 (3)

␦d ij
Si 䡠 Sjd ij0

储␦r i ⫺ ␦r j 储
d ij0

2

parameterized by the spin-phonon coupling
constants ␣ and ␣⬘ and the elastic constants K
and K⬘. The exchange constants now depend on
the deviation of the interatomic distance ␦dijfrom the value d 0ij in the uniform phase. The
atomic displacements ␦ri are free to adjust to
give the best compromise between the increaseof elastic energy and the gain in magnetic energy. The Hamiltonian of Eq. 3 was solved byexact diagonalization of a 16-spin (eight-dimer)
cluster, using the Lanczos algorithm imposing
periodic boundary conditions compatible withthe rhomboid cell and a total magnetization of
1/8, and by minimizing the energy with respect
to ␦ri. Similar calculations have been performed in the context of spin-Peierls systems
such as CuGeO3 (29).
The results of Fig. 3A were obtained assuming typical values ␣ ⫽ ␣⬘ ⫽ 7 and K ⫽K⬘ ⫽ 20,000 K for oxides (27) and using
well-accepted values for the exchange constants, J ⫽ 85 K and J⬘ ⫽ 54 K (30). The
magnetization profile extends over the entire
unit cell, with one strongly polarized dimer
surrounded by decaying oscillation of magnetization. Similar structure has also been
observed around impurities in quantum spin
chains (31, 32) and two-dimensional superconducting cuprates (33). It is analogous to
the Friedel oscillation near impurities in metals. The calculated profile captures the essential feature of the experimental results (Fig.
3B), providing further support for the rhomboid cell. Note that the coupling to phonons

is required merely to select one of the degenerate ground states and is not necessary toobtain a plateau. The magnetization patterndepends only weakly on the value of thespin-phonon coupling (fig. S1); therefore, thequalitative feature mentioned above is expected to be an intrinsic aspect of the magnetic system. Note also that the displacements we obtain (table S2) are very small—less than 1%, which would not be possible todetect experimentally in a magnetic field of
27 T with facilities currently available.
We now discuss the nature of the phasetransition into the 1/8 plateau state. The fielddependence of the magnetization obtained
from the Cu NMR spectra (Fig. 4A) shows anearly discontinuous jump upon entering the
1/8 phase. The vicinity of the phase boundary
was studied by the 11B NMR spectra obtained
at 35 mK (Fig. 4B). The 11B spectra are muchnarrower than the Cu spectra and are easier toobtain. In agreement with the Cu NMR results, the spectrum at 25.93 T with three
sharp peaks split by the quadrupolar interaction indicates a uniform magnetization,
whereas the rich structure of the spectrum at
27.82 T points to a magnetic superlattice. If
the transition were second order and there
were no disorder, we would expect that the
spectral shape changes continuously with thefield. This is, however, not the case: The
spectrum at 26.82 T is a superposition of two
types of spectra, each of which is identical to
the spectrum at 25.93 T or 27.82 T. Thus, thetwo phases coexist at this field. The volume
fraction of the uniform phase, given by theintensity of the sharp central peak, continuously changes in the field range 26.4 to 27 T
without hysteresis (Fig. 4C). Such behavior

Intensity

Intensity

Magnetization <Sz>

Fig. 4. (A) Field depen0.08
dence of the averageAB
magnetization obtained
0.06
from the Cu NMR spectra.
Circles denote data ob0.04
tained at 35 mK; triangles
denote data obtained at
0.5 K. The results below
0.02
26 T correspond to theuniform phase where the0
distribution of Hn has a
16 18 20 22 24 26 28
single peak. The horizonMagnetic field H (T)tal line at 具Sz典 ⫽ 0.0625
25.93 Tshows 1/8 of the satura11
Ction. (B) B NMR spectra
at different magnetic
ﬁelds obtained from sum26.82 T
mation of the Fouriertransformed spectra of
the spin-echo signal at
27.82 T
discrete frequencies. The
frequency is measured
from the unshifted posi26
26.5
27-5 -4 -3 -2 -1 0 1 2 3tion (Hn ⫽ 0) for thequadrupole split central
Frequency (MHz)Magnetic field H (T)line. (C) Field dependence
11of the intensity of the B NMR central line of the uniform-phase spectra measured with increasing(open circles) and decreasing (solid dots) ﬁeld.

398

can be explained if the transition is first orderbut broadened by some disorder in the sample, which enables nucleation of the magnetic
superlattice with small domain size. Interestingly enough, this is consistent with theLandau-Lifshitz theory, according to which
the transition is expected to be first orderwhen the ordering wave vector does not lie
on a high-symmetry point; the ordering wave
vector k ⫽ (␲/a, ␲/2a) for the rhomboid cell
is not a high-symmetry point (27).
Quantum melting of various orderedstructures—for example, vortex lattices in superconductors and charge or orbital order inperovskite transition metal oxides— has been
attracting increasing attention. In many cases,
the phase transition is controlled by chemical
doping, which also introduces disorder andincreases complexity. Here we have a clean
system with an exotic ordered structure, inwhich the phase transition can be tuned by amagnetic field. As the temperature is increased, we also expect a melting transition,
which should be discontinuous for the same
reasons discussed for the field-induced zerotemperature transition (27). This remains to
be studied; at present we only know that the11
B NMR spectrum at 1.5 K shows no signature of superstructure.
References and Notes

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27. See supporting data on Science Online.
28. An alternative way of selecting a unique ground state is
to add a symmetry-breaking external ﬁeld, as is often
done in the simpler case of spin-Peierls transition. In ourcase, however, this would bias the results because we
do not know a priori the magnetization pattern.

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REPORTS
29. A. Dobry, J. A. Riera, Phys. Rev. B 56, R2912 (1997).
30. S. Miyahara, K. Ueda, J. Phys. Soc. Jpn. 69 (suppl. B),
72 (2000).
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34. We thank J. L. Gavilano and P. Van der Linden for help

with the very low temperature measurements, K. Ueda
and M. Imada for useful discussions, and M.-H. Julien
and Y. Tokunaga for technical assistance. M.T. and K.K.
thank Grenoble High Magnetic Field Laboratory for their
hospitality during the course of these experiments. Supported in part by Grants-in-Aid for Scientiﬁc Researchfrom the Japan Society for the Promotion of Science;
the Ministry of Education, Science and Culture, Japan;
and the Swiss National Fund.

Stimulated Raman Scattering in
Hydrogen-Filled Hollow-Core
Photonic Crystal Fiber

Supporting Online Materialwww.sciencemag.org/cgi/content/full/298/5592/395/
DC1
SOM Text
Fig. S1
Tables S1 and S2
13 June 2002; accepted 2 September 2002

ser beam with conventional optics produces
approximately constant high intensity over a
distance equal to twice the Rayleigh lengthzR ⫽ ␲wo2/␭, where wo is half the beamwidth at the focus and ␭ is the vacuum wavelength; in this case, fom ⬃ 16. In a fiber
capillary, the figure of merit takes the form

F. Benabid,* J. C. Knight, G. Antonopoulos, P. St. J. Russell
We report on stimulated Raman scattering in an approximately 1-meter-longhollow-core photonic crystal ﬁber ﬁlled with hydrogen gas under pressure. Light
was guided and conﬁned in the 15-micrometer-diameter hollow core by a twodimensional photonic bandgap. Using a pulsed laser source ( pulse duration, 6
nanoseconds; wavelength, 532 nanometers), the threshold for Stokes (longer wavelength) generation was observed at pulse energies as low as 800 ⫾ 200 nanojoules,
followed by a coherent anti-Stokes (shorter wavelength) generation threshold at
3.4 ⫾ 0.7 microjoules. The pump-to-Stokes conversion efﬁciency was 30 ⫾ 3% at
a pulse energy of only 4.5 microjoules. These energies are almost two orders ofmagnitude lower than any other reported energy, moving gas-based nonlinear
optics to previously inaccessible parameter regimes of high intensity and longinteraction length.
A long-standing challenge in nonlinear optics
is the maximization of nonlinear interactions
between laser light and low-density media
such as gases. The requirements for efficientnonlinear processes are high intensity at low
power, a long interaction length, and a goodquality transverse beam profile. A conceptual
structure capable of delivering all these requirements simultaneously is a perfectly
guiding, hollow-core waveguide supporting asingle transverse mode with low attenuation
losses. Theoretically, this could be realized
by using a perfect metal. However, the attenuation in real metals at optical frequencies is
too high. We report here the use of a hollowcore photonic crystal fiber (HC-PCF) (1) toachieve efficient stimulated Raman scattering(SRS) in hydrogen gas.
Unlike in hollow fiber capillaries, light istrapped in an HC-PCF by a two-dimensional
photonic band gap (PBG) created by a “photonic crystal” of microcapillaries filling theregion around the hollow core (2). Although
theory shows that light can be guided in asingle transverse mode without loss, typicalcurrent HC-PCFs display attenuation losseson the order 1 dB/m. Such low attenuation,
allied with a very small core area (⬃100␮m2 ), make HC-PCF a promising “host” forRaman active gases, offering ultralong interaction lengths while keeping the laser beamtightly confined in a single mode.
A number of conventional approaches
have been used to enhance SRS in gases.
These include focusing a laser beam into thegas with suitable optics, using a ⬃200-␮mbore fiber capillary to confine the gas andprovide some degree of guidance for the light(3), and employing a gas-filled high-finesse
Fabry-Pérot cavity to increase the interaction
length (4). It is instructive to compare thesewith HC-PCF, using the figure of merit fom ⫽Lint␭/Aeff , where Lint is the effective constantintensity interaction length and Aeff is theeffective cross-sectional area. Focusing a la-ƒom ⫽6.8a(n 2 ⫺ 1)␭␲ 冑n 2 ⫹ 1(1)where n is the refractive index of the glass,and we used the results reported in (5). This
increases in a linear fashion with bore radiusa, indicating that small-core capillaries makepoor waveguides. For a hollow-core PCF, thefigure of merit takes the form
ƒom ⫽

␭
␲a 2␣

(2)

where ␣ is the exponential attenuation rate of
the intensity. These three expressions areplotted (Fig. 1) for silica glass (n ⫽ 1.46) ata wavelength of 532 nm.
The HC-PCF has clear advantages, particularly when the bore is small. At a diameterof 10 ␮m, for example, a free-space beam ispreferable to a capillary, whereas an HC-PCF
with a loss of 0.3 dB/m is almost 10,000
times more effective. Improvements in allsorts of nonlinear laser-gas interactions
should become possible, such as ultralowthreshold SRS in gases.
SRS is a two-photon linear inelastic lightscattering process, in which an incoming
photon interacts with a coherently excitedstate of the system (in our case, one of thevibrations of a hydrogen molecule). As a
Fig. 1. Figures of meritfor a hollow capillary
and two HC-PCFs, relative to the value for afree-space beam. At abore radius of 5 ␮m,
the HC-PCF is almost
10,000 times better.

Optoelectronics Group, Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK.
*To whom correspondence should be addressed. Email: pysab@bath.ac.uk

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