Neutron scattering studies onfrustrated spin systems
Masaki Fujita

Topics:● Novel excited state in geometrically frustrated magnet● High-magnetic-field measurement on triangular lattice magnet

Introduction 1

What is geometric frustration?
On regular triangle

On regular tetrahedron

B
JBB: AF

JBB

B
JBB
B

JBB

JBB

?
B

B

?

JBB
JBB

JBB
B

JBB

B

?

No classic spin arrangement
Spin frustration ≡Many degenerated spin arrangements, generated
by such cancellation of exchange interactions

G. H. Wannier, Phys. Rev. 79, 357 (1950). P. W. Anderson, Phys. Rev. 102, 1008 (1956).

Introduction 1
spinel AB2O4

A

Real materials
C15-Laves phase AB2

[111]

B
O

pyrochlore lattice (Corner sharing tetrahedra)

Today’s topics, ACr2O4, LiV2O4,YMn2 etc.

... -kagome-trianglekagome-triangle- ...

Ideal geometrically frustrated lattice, including pyrochlore A2B2O7 also.
Long-range magnetic order can take place with lattice distortion.
Y. Horibe et al. PRL 96, 086406 (2006).

eragedahedraof theccountmetricof sixpy thetetrae foursquarean beðT Þ21 :a wereng theblades,is closegon spin
e
al

$

"#

3'

connects every third nearest neighboring Cr ions that are
exhibits strikingly different behaviors in the tetragonal
k with T, which indicates gradualdisintegration of the directors.
phase below TN % 7:8 K. It undergoes an elongation along
separated by the symmetrically equivalent distance of#12 ; 12 ; 0$. These are the second nearest neighbors along
the c axis [Figs.
and 2(d)] andthe magneticas
longtherange
What is the basis for the emergenceof 2(b)spin-loop
directors
order has an incommensurate characteristic wave vector
the chains represented by lines in Figs. 3(c) and 3(d).
effective degrees of freedom in [seethisFig.frustrated
magnet?
Fig. 4 shows,-'.-' RSTUV'2(c)].
Within the sublattice, spins are aligned according to Sj %The positions
the magnetic BraggreflectionsSpin
found inthe spins surrounding a hexagon
in theof pyrochlore
lattice.
So e2"iQ (#r &r $ and therefore the neighbor spins rotate by'
the scattering plane are shown in Fig. 3(a). The possible
2#, ", and 2# ' " in the ab, ac, and bc plane, respecconfigurations that satisfy all interactions
are characterized
110thecharacteristic incommensurate
wave vectors by
are
QM %
CdFe
2O4 FYZ$[4\3>]1^_`M:aGbcOd-e/
RSTUVWXP
tively, where 2# % !" % 16:2" . In order to construct the fg!; 1$ or #0; 1; !$theor #1;!; 0$ with
! % 0:0894#3$. (hconnected vectors of Fig. 4b.#0;Althoughspin
configuration
relativeorientation between the different sublattices, the011 Fv
h4iidUVjkFlSUVmnFopqa>r1234:sFaRPtuqO

Introduction 1

M

j

0

Paramagnetic scattering of d-el. systems
ZnCr2O4

and k are interchangeable.) However, since c > a, the three……..VWX
magnetic interactions in the tetragonal phase of CdCr2 O4
possible QM ’s would produce
the magnetic Bragg reflecwxy-z*-{|-}w*>r1234:m3YZU~UVO•cvwxy-z*-k
are considered. Analysis of a series of chromium oxides
tions at slightly different positions in the scattering plane.
f€•1234:?‚m3mnO'2&)-345')-345'6*FQMƒf<„1234:'
We performed elastic scanswith a high Q resolution over a
Q&_K`a567;</=>&NbcL4de1fgWKL4567I
set of three IC positions 'to distinguish between the differ-

CdCr2O4

CdFe2O4

Small
hexagon!"#$%&'"()*+,-./.0123+,456789:;/ (.01<(d))/=!">?@AZnFe2O4"BCDEDF?GH(.01 2 (b))"IJKLMN*OP,456
FIG. 2 (color online).

(a) Bulk susceptibility, $, as a functionof T. The transition is of first-order and has a small but finite
QRSTU67

FIG. 1 (color online). Color image of inelastic neutron scattering intensities from single crystals of CdCr2 O4 at T % 15 K >TN with @! % 0:6 meV. The data were taken at SPINS using 11
2:1 ) 15 cm PG(002) analyzer blades with Ef % 5 meV.

ZnFe2O4'

hysteresis over 0.1 degrees. The inset shows a linear fit to 1=$.
(b) T dependence of the normalized integrated neutron scattering
intensity at the magnetic peaks and that of the lattice strains,%a;c % "#a;c$#a;c$ . The data were obtained from fitting the data in (c)and (d) to Gaussian peaks. (c), (d) Elastic neutron scattering data(c) through aGbc'
magnetic #1; !; 0$ IC points and (d) through anuclear (400) Bragg reflection below and above TN .

efgW

Large
hexagon247204-2
a ,b…OddiUVjf†h4d-e/ PF !" = 1 meV F‡lSYZU~UV€•>rh 88BijS7&1klmB1`n#;<opqrsBJ&tKuv6wxyqr
1234:ˆ‰fYZU~UVO•cvwxy-z*-kf€•1234:?‚m3mn
Yz0`7{|MBC7I}~&{|M1•€•jS7wxyM&Yz&‚ƒBJ&{‚' 2&)-345')-345'6*f<„1234:sFŠ‹OŒf 7 •Ž(F•T>„•1M ZnFe2O4 5
MK„…4567;</?~†‡45ˆ‰4567I{|MKn.n.DŠ‹'Œ•K„…G
O‘’^“”•2gƒt–1— 8 •Ž(YS•TF“”4 ZnCr2O4 "4+",+F€•˜™*-f
7Ž6;</`•BC7I8&{|M1•6K‘’4d“#”`:`6•–K„…GH
š234:›9œ7t?@ABCDE.mYS$” ZnFe2O4 f•12 ZnCr2O4 FžŸf'efg—&•–K`7I-.¡mYS$fŠ¢12gƒt£¤¥¦§.s98F' -¨f/01234s5>©ª«4:¬A-•BCDE>®12?@ABCDE.¯;$f”4 &'( °±f…²F³T>‚² ZnCr2O4 5O´3tZnFe2O4 [3][7]POµƒF¶·F¸¹°±.º»234 )g °±f‚c²F³T>¼²MLtFe3+Ž½->¼²'¾x¿%À&Ž(POÁ3˜'FBCDE 70!

Figure 3 Wavevector dependence of the inelastic neutron scatteringcross-section for!"#$!
./0«4F.–1—ÂÃPQ4:›FMLYSŽ½-F´ŸCr3+ Ž½-FÄ¢5
ZnCr2O4. a,b, Colour images ofS.-H.
inelastic Lee
neutronetscattering
intensities
fromsinglecrystalsal., Nature
418,
856(2002). J.-H. Chung et al.,PRL 95, 247204 (2005).
110
CdFe2O4 FYZU~UV€•.š234s5OÅªÆ3ÇWPQ4:.012
of ZnCr2O4 in the (hk0) and (hkk)planes obtained
at T ¼PRB
15 K for70,\q ¼024418
1 meV:
K.symmetry
Kamazawa
et al.,(2004). K. Kamazawa, Ph.D thesis (2004).
The data are a measure of the dynamic form factor for self-organized nanometre-scale
V"WX"YZ[\R3.01]3+^7-_,`a3L6!Q*bc6-d3#$%/',3
spin clusters in the material. c,d, Colour images of the form factor
squared calculatedfor
antiferromagnetic hexagon spin loops averaged over the four hexagon orientationsin thee54f54567

neighbor spins. For comparison the dashed line shows thecluster form factor. As opposed to thetheparamagnetic
form factor
for Figs.
an atomic
phase,
1(a) and 1(b) provide
80DTU7VQVWXXI8&Q1YKZ[/\;</]K^S7MNOPBC7
Thermaland quantum
fluctuations that violate collinearity
magnetic neutron scattering intensity forevidenceforquantum
criticalfluctuations
of
small
AFM
spin, the cluster form factor vanishes for Q ! 0, which is evidence within thepowder-averaged
induceinteractions between
anhexagons
isolated spinshould
dimer at thenearestresidual
neighbor separation
clusters most likely antiferromagnetically correlated
that clusters carry no net spin11. Further
analysisiscomplicatedby
r0 ! 2.939
Å [15]. The
spinaccountpair modelatetrahedra. The data closely resemble those obtainedindirectors.
Suchinteractions
may
forproduces
the inelasticityof thbroader
peak
than
the
experimentindicatingthatthesimilarexperimentsonotherfrustrated
AFM’s[7,14].
the anisotropy of spin clusters, which can occur in four different scattering, the director correlations reflected in the greater sharp
resonating spin cluster in ZnCr2 O4 is more complex.
For T , Tc , however, the low energy spectral weightorientations for a cubic crystal. Rather
than Fourier-inverting the2(b) also shows
the wavevector2 dependenceof the increase oof the Figure
experimental
featuresin Figs
and 3, andconcentrates into a sharp constant-energy modenesscentereddata, we therefore compare them tothe! orientationallyinelastic
scattering
below
the
resonance
integratedover
at h̄v
4.5 meV # jJj ¿ kBaveraged
Tc . Above the kresonancewith T, whichindicates gradual disintegration of the directors.
energy from 1 to 3 meV (open symbols). Weak, nonis an Individual
additional bandof intensity centered at
Fourier transform of potential spinthere
clusters.
tetrahedra……..VWX
the basis for theof spin-loop
as thpeaksemergence
are visible andtheir locationsdirectors
h̄v # 9 meV [Fig. 2(a)]. Careful inspection ofWhat
data is resolution-limited
would be prime candidates, as they constitute
the
basic
motif
of
the
coincide
with
AFM
Bragg
peaks
(shaded).
Excitations
below the resonance also reveals weak dispersing
streaks degrees of freedom in this frustrated magnet? Fig. 4 show
effective
below
the resonance are also apparent in the wave vector
emanating isfrom
AFM
Braggtopoints.
pyrochlore lattice. However, a tetrahedron
too
small
account the spins surrounding
Q&_K`a567;</=>&NbcL4de1fgWKL4567I
a hexagon in the pyrochlore lattice. Spinintegrated data of Fig. 2(a) where the intensity increases
Focusing
first
on
the
orderedphase,
Fig.
2(a)
shows
for the features observed (Fig. 3a, b).
Thenext-smallest
symmetric
configurations
that satisfy
all forinteractionsare(thecharacterized
by thin proportion
to energyh̄v , 3.5 meV
low
the Q-integrated
at
2 4 magnetic scattering cross section
structural unit is the hexagonal loop
formedbyaclusterofsixenergyupturnofis incompletely
resolved elasticthe
scattering).
T ! 1.7 K. With a resolution corrected FWHM
ofconnected
vectors
Fig. 4b. Althoughspin configuration
Bothfeaturesareconsistentwithneutronscattering
from
only
0.8
meV
the
h̄v
!
4.5
meV
peak
is
evidence
tetrahedra (Fig. 4a). Two spins from each tetrahedron occupy thespin waves in a three dimensional AFM powder with avertices of a hexagon, while the other two spins from each tetraspin gap D , 1.5 meV. In particular, the ratio of elastic
to inelastic scattering is consistent with estimates based
hedron belong to different hexagons. Averaging over the fouron spin wave theory in the long wavelength limit. Fromdifferent hexagon orientations in the pyrochlore lattice, the square
the width of the peaks we estimate a spin wave velocityof the antiferromagnetic hexagon
spin-loop
form
factorcanbepwd
sxy ! 18!2" meV Å. This number is much less than the

Introduction 2(a) Expanded geometric frustration?
ZnCr O•in magnetically ordered phase

Para phasequasielastic scatt.
TN=13 K

efgW

h 88BijS7&1klmB1`n#;<opqrsBJ&tKuv6wxyq

Yz0`7{|MBC7I}~&{|M1•€•jS7wxyM&Yz&‚ƒBJ&two discrete modesMK„…4567;</?~†‡45ˆ‰4567I{|MKn.n.DŠ‹'Œ•K„…

AF phase
inelastic scatt.

7Ž6;</`•BC7I8&{|M1•6K‘’4d“#”`:`6•–K„…G

at 4.5 & 9.0 meV

efg—&•–K`7I

FIG. 2. Integrated magnetic scattering intensities at T !
1.7 K derived from Fig. 1(c). R(a) v dependenceR of theQ-integrated intensity: Ĩ!v" ! Q 2 dQ Ĩ!Q, v"$ Q 2 dQ.
The horizontal bar shows the instrumental energy resolution.
70( b) Closed symbols show the Q dependence
R 6.0-meV of the v-integrated
FIG. 1 (color). Color contour maps of the magnetic neutron
resonance intensity: Ĩ!Q" ! h̄ 3.5-meV Ĩ!Q, v" dv. OpenS.-H.
Lee et al., PRL 84, 3718 (2000).
circles show data integrated from 1 to 3 meV. The solid line
scattering intensity versus wave vector and energy transshows the elastic scattering cross section scaled by a factor offer at three temperatures spanning the phase transition at
10. Shaded peaks are magnetic Bragg peaks.
Tc ! 12.5!5" K.

Unknown modes!

3719

Results on MgCr2O4

PWD inelastic scatt. data
ordered phase
a

0.05

0.10

0.15

b

0.20

25

5K

25

15
10

2nd mode5
1st mode

20

5K

Energy [meV]Counts / meV

20

0.2

15
10

0.1

5

0
0.0

0.5

1.0

1.5

2.0

2.5

Wave number Q [1/Å]discrete modes.

0.10

0.15

0.20

25
0.3

5K

0.05

c

30

Intensity [a.u.]

Energy [meV]

para phase

0 0.0
-10

0

10

20

0Energy5 transfer
10 [meV]
15

Energy [meV]

30

20

20

15 K

15
10
5
0
0.0

0.5

1.0

1.5

2.0

2.5

Wave number Q [1/Å]

quasielastic scatt.

the
spin
omic
ence
d by
rent
the
aged
edra
f the
ount
etric
f six
the
etrafouruare
n bedegrees of freedom for the pyrochlore lattice, we call the corresponding low-temperature state of the frustrated magnet a protectorate.)
Thermal and quantum fluctuations that violate collinearity
within the hexagons should induce residual interactions between
directors. Such interactions may account
2 for
4 the inelasticity of thescattering, the director correlations reflected in the greater sharpness of the1st
experimental
mode features in Figs 2 and 3, and the increase of
k with T, which indicates gradual disintegration of the directors.
What is the basis for the emergence of spin-loop directors as theeffective degrees of freedom in this frustrated magnet? Fig. 4 shows
the spins surrounding a hexagon in the pyrochlore lattice. Spinconfigurations that satisfy all interactions are characterized by theconnected vectors of Fig. 4b. Although the spin configuration

Results on MgCr O

SX inelastic scatt. data

J. Phys. Soc. Jpn., Vol. 71, No. 2, February, 2002same

2nd mode

Y. YASUI et al.

601

similarFig. 2. Figure 3 shows the energy profiles taken at Q ¼
ð0; 0; 2:1Þ at several temperatures. In both figures, we can seethat the quasi elastic scattering component with asymmetricshape with respect to the E ¼ 0 point, is superposed onto theelastic component which has the Gaussian profile. (Theasymmetry is just due to the Bose factor.) The intensity ofthe elastic scattering depends on the temperature and thescattering vector Q. It decreases with increasing magnetic
field H,15) indicating that the magnetic scattering contribution exists in the elastic scattering.
Figures 4(a) and 4(b) show the energy scan profiles
obtained at several temperatures with !E ¼ 81 !eV at Q ¼
ð1:25; 1:25; 0Þ and (1:9; 1:9; 1:9), respectively. Well-definedmagnetic excitations are observed at the energy region of 1–

6 spinparamagnetic data cluster
on Zn/MgCr2O4
S.-H. Lee et al., Nature 418, 856 (2002).

7 spin paramagnetic datacluster
on Tb2Ti2O7
Fig. 5. Map of the scattering intensity observed in the (h; h; l) plane in thereciprocal space at 0.4 K with the neutron transfer energy E ¼ 0 meV and!E ¼ 81 !eV.

meV. With decreasing T, the softening and the hardening
Y. Yasui et of2al.,
JPSJ 71, 599 (2002).
the excitation take place at around Q ¼ ð1:25; 1:25; 0Þ
and (1:9; 1:9; 1:9), respectively, and the energy dispersion
becomes significant. We have now found that the magnetic
scattering has all the elastic, quasi elastic and inelastic
contributions.
Figure 5 shows the scattering intensity map taken in the(h; h; l) plane in the reciprocal space at 0.4 K and at
E ¼ 0 meV with !E ¼ 81 !eV. The nuclear Bragg peaks

Results on MgCr2O4

SX inelastic scatt. data
1st mode

2nd modeResults on MgCr2O4

Summary of results

a

0.05

0.10

0.15

0.20

b

10

5K

Energy [meV]Intensity [a.u.]

Energy [meV]

0.3

5K

15

0.2

0.1

5
0
0.0

0.10

0.15

0.20

25

25
20

0.05

c

0.5

1.0

1.5

2.0

Wave number Q [1/Å]para & 1st mode

2.5

0.0

0

5

10

15

Energy [meV]

20

20

15 K

15
10
5
0
0.0

0.5

1.0

1.5

2.0

2.5

Wave number Q [1/Å]

2nd mode

Universarity of spin molecule and the origin?

High-magnetic-field measurementon triangular lattice magnet
principal investigator
Dr. K. Ohoyama

collaborators:
T. Yoshii,
Y. H. Matsuda
H. Nojiri

The Present situation ofMagnetic Field for ND
The Present Situation of Neutron Diffraction
SC magnet：

H<17.5T(Germany, HMI)
H<13T(Japan, JAEA)
Pulsed Magnet: H<25T (in KEK)
（Motokawa et al, Mitsuda et al.)

ND experiments over 10T are
still difficult and limited for thepresent.
Easier and more Diffusivetechniques are ambitioned.
/42~4m
The Condenser Bank used in KEK(It’s too large!)

Solenoid Coil Magnet
Coil Parameter
ID=12 mm, OD~33mm, length=16 mm filling factor 65 %Wire 1 mm diameter Cu-Ag round wire
Resistance 82 % of Cu at R.T. R77K/RRT=0.27~14 turn/layer, 10 layer
L=0.226 mH, R77K=73 mOhm

H

An example wave form with our Capacitor bank of C=5.6 mF,
L=8 microF, R=40 mOhm and charging voltage of 1.5 kV
30 T at 4.6 kA

33mm

32T

2msec

SUS support pipe

/42G10-support plate

Co-axial tube

MagnetCoil Cooling System
Interval is dependent on the cooling power.
Orange Cryo

Higher Cooling Power
Coil : Liquid N2
Sample : Liquid He

L N2

He gasShortening of pulse interval

L He(~3-5min.)磁場

2msec/5min~0.001%Beam

Coil~φ7mm×15mm �

Insert2θ<30°

8°
22°

/42

Instruments
Tohoku Univ. Neutron Spectrometer AKANE
Compact Experiments!

@ JRR-3M, JAEA（Tokai, Japan）

AKANE

Controller Unit

Condensers
5.4mF
3kV
25kJ

Condenser
Bank

Hmax~40T/42

This system can be installed atany spectrometers in Japanand oversea Facilities.

Results: CuFeO2 : ND on AKANE (H=0T)
Single Crystal ( Mitsuda )φ3.5mm 5mm

4-Sub. Peak at (1/4,1/4,3/2)
H=0T
AKAEN 19.7meV g-O-S-B-B 2axis 1Det
5000

CNT / 2sec

4000

(1/4,1/4,3/2)

3000

h-scanΔQ/
Q=1.2%
S/N~150

T=5.0K
T=22.8K

2000
1000
0

0.23

0.24

0.25

0.26

0.27

(h,h,3/2)
c-axis[110]/42

We measured the time dependenceof the peak top intensity undermagnetic field.

Disappearance of 4-Sub. Peak under H=10T
Peak Top Intensity of 4-Sub. Peak
AKAEN 19.7meV g-O-S-B-B 2axis 1Det
At (1/4,1/4,3/2)
12

4-Sub
60

7T

40

4
20
2

6h20m
-2

0

2

4

6

(1/4,1/4,3/2) T=5K

CNT / 160µsec/100pulse

30

8

0

100pulse, 6.5hr

7T

25
20
15
10
5
0

/42

6

Time (msec)(1/4,1/4,3/2)

10
8

0
-4

4-Sub. Peak disappearsfor H>7T.

(1/4,1/4,3/2)
T=5.0K

0

Magnetic Field (T)CNT / 400µsec/100pulse

80

2

4

H (T)

6

8

10

31.5T Experiments: World Record for ND!
AKAEN 19.7meV g-O-S-B-B 2axis 1Det

CNT/ 80µsec/ 136 pulse

30
25

40
5-Sub

(0.207,0.207,3/2)
T=4.9K30

20

136pulse
15hr.

15
10

20
10

5
0

0

1

2

3

4

5

Magnetic Field (T)5-Sub. Peak0

Time (msec)(0.207,0.207,3/2)

5-Sub.
(0.207,0.207,3/2)/42

CNT/ 160µsec/ 136 pulse

255-Sub.
T=4.9K

20
15
10
5
0

0

5

10

15

20

25

30

Magnetic Field (T)31.5T is the highest magnetic fieldin the World for ND.
40T

3-Sub. Peak under 29T

Neutron Counts at (1/3,1/3,2)
AKAEN 19.7meV g-O-S-B-B 2axis 1Det
14

CNT/ 160µsec/ 146pulse

3-Sub

12

(1/3,1/3,2)
T=4.9K

1030
25146pulse
15.5hr.

8

20

6

15

4

10

2

5

0

0

1

2

3

4

5

Magnetic Field (T)22T<H<32T

35

0Time (msec)(1/3,1/3,2)

CNT/ 160µsec/ 146pulse

15

CuFeO2
T=5.0K
10 (1/3,1/3,2)

22Ｔ

5

0
0

5

10

15

20

25

30

Magnetioc Field (T)3-Sub. Structure/423-Sub. Structure is induced above 20T.

Summary
Novel excited state in geometrically frustrated magnet• Nature of frustration is quite robust
as molecular spin excitations were observed in the ordered phase.
Study for the universality of dynamic geometric frustration andthe origin of spin molecules is important.

High-magnetic-field measurement on triangular lattice magnet• Compact system of magnetic field generation
for neutron scattering experiment was developed.

We succeeded in observing change of magnetic reflections in
CuFeO2 under 31.5T, the highest magnetic field for ND experiments.