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arXiv:cond-mat/0505066v1 [cond-mat.other] 3 May 2005

Magnetic excitations in {Mo72Fe30}

V. O. Garlea,1, ∗S. E. Nagler,2J. L. Zarestky,1C. Stassis,1D. Vaknin,1P. K¨ ogerler,1D. F. McMorrow,3,4

C. Niedermayer,5D. A. Tennant,6B. Lake,7,1Y. Qiu,8M. Exler,9J. Schnack,9and M. Luban1

1Ames Laboratory, Department of Physics and Astronomy, Iowa State University, Ames, IA, 50011, USA

2Oak Ridge National Laboratory, Oak Ridge, TN 37831 USA

3Risø National Laboratory, DK-4000, Roskilde, Denmark

4Department of Physics and Astronomy, University College London, UK

5Laboratory for Neutron Scattering ETHZ & PSI, CH-5232 Villigen PSI, Switzerland

6School of Physics and Astronomy, University of St Andrews, St Andrews, FIFE KY16 9SS, Scotland, UK

7Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK

8NIST Center for Neutron Research, Gaithersburg, MD & U. Maryland, College Park, MD, USA

9Universit¨ at Osnabr¨ uck, Fachbereich Physik - D-49069 Osnabr¨ uck, Germany

(Dated: February 2, 2008)

We report cold-neutron inelastic neutron scattering measurements on deuterated samples of the

giant polyoxomolybdate magnetic molecule {Mo72Fe30}. The 30 s = 5/2 Fe3+ions occupy the

vertices of an icosidodecahedron, and interact via antiferromagnetic nearest neighbor coupling. The

measurements reveal a band of magnetic excitations near E ≈ 0.6 meV. The spectrum broadens and

shifts to lower energy as the temperature is increased, and also is strongly affected by magnetic fields.

The results can be interpreted within the context of an effective three-sublattice spin Hamiltonian.

PACS numbers: 75.25.+z, 75.50.Ee, 75.75.+a, 78.70.Nx

Magnetic molecules are ideal prototypical systems for

the study of fundamental problems in magnetism on the

nanoscale level [1]. As a result, their properties have

been the subject of many theoretical and experimental

investigations [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

paramount importance is the determination of the mag-

netic excitation spectrum. However, for a molecule with

N magnetic ions of spin s the calculation of the (2s+1)N

eigenstates and their energies quickly becomes impracti-

cal for increasing N and s. Neutron scattering is the

most effective and direct technique for determining the

magnetic energy levels, and there have been studies of

excitations in several magnetic molecules containing up

to 12 spins [2, 3, 4, 5, 6, 7].

In this Letter we report cold-neutron inelastic scat-

tering results obtained on one of the largest magnetic

molecules yet synthesized: the polyoxomolybdate cluster

{Mo72Fe30}. The crystallographic structure is described

by the space group R3 with the lattice constants: a ≈

55.13˚ A, and c ≈ 60.19˚ A [9]. The molecule contains

30 Fe3+ions (s = 5/2) occupying the vertices of an

icosidodecahedron. The magnetic ions are interlinked by

Mo6O21 fragments acting as super-exchange pathways,

resulting in nearest neighbor antiferromagnetic exchange

J?Si·?Sj and a singlet spin ground state. As the icosi-

dodecahedron consists of 20 corner-sharing triangles cir-

cumscribing 12 pentagons the spins are frustrated and

show properties similar to the antiferromagnetic Kagom´ e

lattice [13]. For s = 5/2 it is reasonable to consider clas-

sical spin vectors as a starting point [8], leading to a

picture at T = 0 of three sublattices of 10 parallel spins

each, with orientations defined by coplanar vectors offset

by 120◦angles. As discussed below, many features of the

observed scattering can be interpreted in the context of

Of

a solvable three-sublattice effective Hamiltonian substi-

tuted for the intractable Heisenberg Hamiltonian [8, 10].

Most of the neutron scattering experiments were per-

formed on deuterated samples to minimize the attenua-

tion and incoherent scattering from the hydrogen atoms.

Characterization of the deuterated samples by infrared

and Raman spectroscopy and X-ray diffraction confirmed

that their properties were consistent with those of non-

deuterated samples studied earlier [9].

The neutron measurements used polycrystalline sam-

ples of approximately 10 g sealed in copper holders under

He atmosphere. Preliminary characterization by powder

diffraction was performed at the HB1A instrument at

HFIR. Upon cooling from room temperature the diffrac-

tion patterns in both deuterated and non-deuterated

samples exhibited a remarkable increase in the back-

ground over the entire measured range of scattering an-

gle. The presence of this scattering at large wavevectors

is a clear indication that it is not magnetic in origin. It

can be understood as arising from quenched static struc-

tural disorder, and is manifested as a very large zero en-

ergy peak in the inelastic scattering spectra. It presents

a significant experimental challenge, and prevents a clean

observation of low energy magnetic scattering.

The inelastic neutron spectra were collected at low

temperatures using three different spectrometers: RITA2

at PSI, DCS at NIST, OSIRIS at ISIS. The data pre-

sented here were obtained at OSIRIS using a fixed fi-

nal neutron energy Ef = 1.845 meV. The results ob-

tained with DCS and RITA2 are consistent with those of

OSIRIS. A complete description of all of the results will

be presented elsewhere [14].

Typical data obtained are shown in Fig. 1. The scans

plotted in all figures are integrated over the range Q =

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FIG. 1: (a) Inelastic neutron scattering spectrum at a nom-

inal temperature of 65 mK. The intensity axis is on a log-

arithmic scale. The solid line shows a best estimate of the

non-magnetic background (see text). Inset: The background

subtracted scattering. (b) Raw data for several different tem-

peratures plotted on a linear scale.

0.9 to 1.8˚ A−1. Positive energies correspond to neutron

energy loss. The upper panel (Fig. 1(a)) shows a semi-

log plot of the spectrum at the base temperature, nom-

inally T= 65 mK. The large elastic peak has a FWHM

of 0.021(1) meV consistent with the instrumental energy

resolution. A much broader peak is visible as a shoul-

der in the plot. The lower panel (Fig. 1(b)) shows the

scattering on a linear scale at four different temperatures

ranging up to 6.5 K. At base temperature a clear peak

is evident near an energy transfer of 0.6 meV. As the

temperature is raised this peak weakens, and there is a

notable increase in the intensity of the scattering at low

energies. This behavior of the intensity is expected if the

peak at 0.6 meV is due to a magnetic transition from

the ground state.The temperature-enhanced low en-

ergy signal could in principle arise from lattice vibrations

or from magnetic scattering, and differentiating between

these possibilities requires careful analysis.

Cooling powders to mK temperatures is known to be

extremely difficult, and since the temperature is mea-

sured external to the sample it is desirable to have an

internal consistency check. This can be accomplished in

principle by checking the detailed balance condition, but

ambiguities in the scattering background, limited data for

neutron energy gain, and a resolution dependent on the

energy transfer limit the precision that can be achieved.

Going through this exercise suggests that the nominal

sample temperatures at 1.8 K and above are probably

reliable, but the base temperature to which the powder

is actually cooled is less certain. The evolution of the

scattering shows that it is clearly well below 1.8 K but it

is likely warmer than the nominal value 65 mK. However,

our data analysis does not depend on the precise value

of the base temperature.

To extract more quantitative information the base tem-

perature scattering was fit to a simple model. It was

assumed that the instrumental resolution at the elastic

position can be described by the sum of two co-centered

peaks: a dominant Gaussian and a Lorentzian with the

latter accounting for the extended tails. The nonmag-

netic background scattering is taken as the sum of an

elastic term proportional to the resolution function plus

an energy independent constant. The magnetic scatter-

ing was assumed to be represented by a single peak,

with a Gaussian found to provide a better fit than a

Lorentzian. Least squares fitting of this simple model

to the data was carried out initially using the entire data

set. Alternatively, the analysis was repeated by deter-

mining the model background alone by fitting to the data

excluding a range in the vicinity of the inelastic peak.

Subsequently a single magnetic peak was fitted to the

data with the above determined background subtracted.

The results were found to be independent of the size of

the excluded range and consistent with the fit over the

whole range. The solid curve in Fig. 1(a) illustrates the

background, and the inset shows the background sub-

tracted data fit to a single Gaussian with peak position

0.56(1) meV and FWHM 0.66(1) meV. The numbers in

parentheses are the estimated uncertainties in the last

digit, and account for both fitting errors and systematic

effects arising from different background estimations. As

seen in the inset to Fig. 1(a) the overall distribution is

modeled well by a single peak, however the data shows

additional structure, including one or more shoulders on

the low energy side. The observed energy width of the

shoulders and the overall distribution are intrinsic since

the resolution is roughly FWHM 0.02 meV. The signal

for energy transfers greater than 0.2 meV is independent

of the assumptions used in determining the background.

Further analysis was carried out to gain more insight

into the temperature dependence of the scattering. A

thorough attempt was made to fit the data for energy

transfers less than 0.5 meV and temperatures up to 6.5

K to models of both single and multiple phonon scat-

tering from solids or fluids [15]. This attempt was un-

successful. Subsequently, the data was examined in a

model-independent fashion, integrating the energy cuts

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FIG. 2:

scattering, plotted with the nonmagnetic background sub-

tracted. Inset: Theoretical scattering for several tempera-

tures, calculated using the quantum rotational band model

with simplifying assumptions (see text).

Temperature dependence of the inelastic neutron

over different wavevector ranges varying from QL= 1.0

– 1.2˚ A−1to QU= 1.6 – 1.8˚ A−1. It was verified that for

all temperatures up to 6.5 K there is no Q dependence

in the scattering to within the sensitivity of the measure-

ment. This is a significant result since the intensity for

single phonon scattering is expected to increase propor-

tional to Q2, and that of multiphonon scattering should

increase even more rapidly with Q [15]. The intensity

of phonon scattering in the range QUshould be stronger

by a factor of 2.5 or more than that in QL, contrary to

the experimental observation. It can be concluded that

phonon scattering is insignificant over the wavevectorand

temperature range of the data considered here.

Given this result, it is reasonable to assume that the

nonmagnetic background is temperature independent.

The background subtracted scattering for several tem-

peratures up to 6.5 K is shown in Fig. 2. With the uncer-

tainties in temperature and instrumental resolution there

may be some amount of magnetic scattering included as

background at low energy transfers, possibly contribut-

ing to an over-subtraction at energy transfers below 0.2

meV. Notwithstanding these ambiguities we believe that

this data provides a fair representation of the inelastic

magnetic scattering in {Mo72Fe30}.

The magnetic nature of the observed spectrum was

confirmed by studying the effect of an externally applied

magnetic field. As illustrated by the base temperature

data in Fig. 3 the application of a field leads to a notice-

able broadening of the peak. With increasing magnetic

field the intensity of the main peak near 0.6 meV de-

creases and the scattering distribution broadens. Exam-

ining the difference between scattering at zero and non-

zero fields [14] shows that the spectrum in a field consists

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FIG. 3: Scattering at base temperature for several different

values of applied magnetic field. Inset: Upper mode energy

as a function of field.

of a central peak with two almost symmetric side bands,

the positions of which vary roughly linearly with field

strength. The position of the central peak shifts slightly

with the applied field; the value at 7 T is estimated as

0.50(1) meV. Following the detailed field dependence of

the lower sideband quantitatively is difficult because of

the large background at low energy transfers. The higher

energy sideband can be observed more cleanly, and the

peak position as a function of field is plotted in the inset

to Fig. 3. For very small fields the side peaks cannot be

resolved, but over the visible range the slope of the upper

peak is 0.049(1) meV/T.

At present there is no rigorous theoretical calculation

available for a detailed comparison with the results of

this experiment. The large number of s = 5/2 magnetic

ions per molecule precludes the diagonalization of the

quantum Heisenberg Hamiltonian. However, the main

features of the energy spectrum can be established using

a solvable effective 3-sublattice Hamiltonian [8, 10] as a

starting point. Defining the total spin on each sublattice

as˜

Sαthe Hamiltonian in the absence an external mag-

netic field is taken asJ

5

˜

SA·˜

This neglects anisotropy, which is estimated to be small

compared to J [11]. The resulting spectrum has a

hierarchy of discrete energy levels for each value of

the total spin quantum number S given by E(S) =

J

10(S(S + 1) − SA(SA+ 1) − SB(SB+ 1) − SC(SC+ 1)),

where the spin quantum numbers span the range 0 ≤ S ≤

75 and 0 ≤ SA,B,C ≤ 25.

form a sequence of well-separated, highly degenerate

“rotational bands” with excitation energies depending

quadratically on S. The energy levels in the lowest two

rotational bands are given as E1(S) =

E2(S) = 5J + E1(S) with degeneracies for S ≤ 50

D1= (2S+1)2and D2= 27(2S+1)2. The corresponding

?

SB+˜

SB·˜

SC+˜

SC·˜

SA

?

.

The low-lying excitations

J

10S (S + 1) and

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levels of the exact Heisenberg model can be expected to

differ by splittings of these levels.

Using the value of J = 1.57 K found to describe the

susceptibility data [9] and the selection rules ∆S,∆M =

0,±1 for neutron scattering leads to an estimate of the

gap between the the ground state and the first excited

state within the band E1(S) as J/5 ≈ 27 µeV. The large

elastic background precludes an observation of this mode

in the present experiment. Using the same energy scale,

transitions from the ground state to the second rotational

band should produce a pronounced intensity near 5J ≈

0.67 meV, consistent with the broad band of excitations

seen at base temperature, as shown in the inset to Fig. 1.

The detailed lineshape of the base temperature scat-

tering may reflect the richer structure of the energy spec-

trum of the nearest-neighbor Heisenberg model, as com-

pared to that of the rotational band model. It may also

reflect the effects of weak anisotropy. A recent approxi-

mate spin-wave calculation [12] for the nearest-neighbor

model with anisotropy predicts that several modes should

be visible in the region of interest. A full understanding

of the line shape requires further theoretical work.

The temperature dependence of the scattering as

shown in Fig. 2 can also be considered within the context

of the rotational band model. As the temperature in-

creases above T ≥ J/5 ≈ 0.3 K, energy levels within the

lowest band become more populated, giving rise to the

broadening and shifting of the main excitation towards E

= 0. A detailed comparison with experiment requires an

evaluation of matrix elements. A calculation was carried

out starting from standard formulas for magnetic neu-

tron scattering [15], using a common nonzero matrix ele-

ment for all allowed transitions [16]. Thermal occupation

of levels was accounted for by a Boltzmann factor, and

Dirac delta function factors associated with allowed tran-

sitions were replaced by a Lorentzian with a width of 0.3

meV. The inset to Fig. 2 shows the resulting calculated

curves using J = 1.57 K. Despite the simplified nature of

the approximations made there is a striking resemblance

between the curves and the data. The peak position of

the theoretical curve at 65 mK is in reasonable agreement

with observations. For successively higher temperatures

the peak broadens and shifts to lower energies. Near 7

K the peak in the intensity occurs at around 0.2 meV,

similar to that seen experimentally. The simplified rota-

tional band model provides a reasonable explanation for

the qualitative behavior with temperature.

An extension of the rotational-band model to non-zero

fields predicts that the ground state gradually shifts to

S > 0 states preserving the quadratic dependence on S in

both the lowest and first rotational bands [10]. Zeeman

splitting of the levels leads to excitations whose energies

vary linearly with the magnetic field. However, a quanti-

tative theoretical explanation of the field dependence of

the neutron scattering cross-section remains as an open

problem.

In summary, using inelastic neutron scattering we have

measured the temperature and magnetic field depen-

dence of magnetic excitations in the Keplerate molec-

ular magnet {Mo72Fe30}.

model [10] accounts for the overall energy scale and qual-

itative temperature dependence of the observed inelastic

scattering. The principal mode observed can be under-

stood as arising from transitions between the two lowest

rotational bands. A quantitative understanding of the

detailed T = 0 lineshape and the behavior in a magnetic

field will require a more sophisticated theory. We hope

that these results will stimulate the continued develop-

ment of theoretical methods incorporating the essential

qualitative features and symmetries of the system, and

that these will prove useful for other systems where di-

agonalization of the Hamiltonian cannot be performed.

Finally we note that the neutron scattering experiments

on large magnetic molecules are currently very difficult,

nonetheless important new information is attainable now,

and next generation instrumentation presents significant

new opportunities.

The authors thank R. J. McQueeney and B. Normand

for helpful discussions. Work at ORNL is supported by

the U.S. DOE under Contract No. DE-AC05-00OR22725

with UT-Battelle, LLC and work at Ames Laboratory

was supported under Contract No. W-7405-Eng-82. The

work at NIST was supported in part by the NSF under

Agreement No. DMR-0086210. The work at Universit¨ at

Osnabr¨ uck was supported by DFG No. SCHN-615/5-2.

The OSIRIS measurements (DAT, BL) were supported

by UK EPSRC GR/N35038/01.

A solvable three-sublattice

∗Electronic address: garleao@ornl.gov

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[12] O. C´ epas and T. Ziman, cond-matt/0412244.

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[14] V. O. Garlea et al. (to be published).

[15] See, e.g. V. F. Turchin, Slow Neutrons, Sivan Press,

Jerusalem (translation), 1965, or S. W. Lovesey, Theory

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