arX

iv:1

402.

0701

v1 [

phys

ics.

atm

-clu

s] 4

Feb

201

4

Spin and orbital magnetic moments of size-selected iron, cobalt, and nickel clusters

and their link to the bulk phase diagrams

A. Langenberg,1, 2 K. Hirsch,1, 2 A. Lawicki,1 V. Zamudio-Bayer,1,3 M. Niemeyer,1, 2

P. Chmiela,1, 2 B. Langbehn,1, 2 A. Terasaki,4, 5 B. v. Issendorff,3 and J. T. Lau1, ∗

1Institut fur Methoden und Instrumentierung der Forschung mit Synchrotronstrahlung,

Helmholtz-Zentrum Berlin fur Materialien und Energie GmbH,

Albert-Einstein-Straße 15, 12489 Berlin, Germany2Institut fur Optik und Atomare Physik, Technische Universitat Berlin, Hardenbergstraße 36, 10623 Berlin, Germany

3Physikalisches Institut, Universitat Freiburg, Stefan-Meier-Straße 21, 79104 Freiburg, Germany4Cluster Research Laboratory, Toyota Technological Institute,

717-86 Futamata, Ichikawa, Chiba 272-0001, Japan5Department of Chemistry, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan

(Dated: February 5, 2014)

Spin and orbital magnetic moments of cationic iron, cobalt, and nickel clusters have been de-termined from x-ray magnetic circular dichroism spectroscopy. In the size regime of n = 10 − 15atoms, these clusters show strong ferromagnetism with maximized spin magnetic moments of 1 µB

per empty 3d state because of completely filled 3d majority spin bands. The only exception is Fe+13where an unusually low average spin magnetic moment of 0.73± 0.12 µB per unoccupied 3d state isdetected; an effect, which is neither observed for Co+13 nor Ni+13. This distinct behavior can be linkedto the existence and accessibility of antiferromagnetic, paramagnetic, or nonmagnetic phases in therespective bulk phase diagrams of iron, cobalt, and nickel. Compared to the experimental data,available density functional theory calculations generally seem to underestimate the spin magneticmoments significantly. In all clusters investigated, the orbital magnetic moment is quenched to5− 25% of the atomic value by the reduced symmetry of the crystal field. The magnetic anisotropyenergy is well below 65 µeV per atom.

PACS numbers: 75.30.Cr, 75.75.-c, 36.40.Cg, 33.15.Kr

I. INTRODUCTION

Only very few elements in the periodic table show mag-netic order at room temperature. In bulk iron, cobalt,and nickel, this magnetic order arises from the balanceof electron localization that gives rise to magnetic mo-ments, and of electron delocalization as a prerequisitefor direct exchange interaction1. Even in these ferromag-netic 3d transition elements, the magnetic moments aresmaller than in the corresponding atoms because elec-tron delocalization reduces the spin magnetic momentto 30 − 50 % of their atomic values, while crystal fieldeffects quench the orbital magnetic moment to less than5 %. The evolution of spin and orbital magnetic momentsfrom atoms via nanoscale matter to the bulk has thereforebeen studied intensively by theory and experiment2–6.In neutral clusters of the ferromagnetic 3d transitionelements iron, cobalt, and nickel7–14, enhanced mag-netic moments and superparamagnetic behavior15 werefound because of narrow 3d bands that are created bythe low coordination of surface atoms in clusters. Fur-thermore, transition elements that show antiferromag-netic order or even no magnetic order in the bulk canexhibit finite magnetic moments in small clusters16–19.Metastable magnetic species of iron and cobalt clus-ters have also been observed20. When deposited on sur-faces, adatoms21–23, clusters24–27, and nanoparticles28–31

were investigated by means of x-ray magnetic circu-lar dichroism spectroscopy24,25,27,31–35, spin polarized

scanning tunneling microscopy36,37 or scanning tunnel-ing spectroscopy38–41, giving access to spin and orbitalmagnetic moments, magnetic anisotropy energies, or ex-change coupling constants of these systems. Large mag-netic anisotropy energies up to 9 meV per atom of ironand cobalt impurities deposited on platinum surfaces21,40

as well as ferromagnetic behavior at room tempera-ture of iron-cobalt nanoparticles on a silicon surface42

have been reported. However, many properties of de-posited clusters and nanoparticles are caused by thestrong interaction with the support that can enhance oreven quench magnetic moments33, and that also gov-erns the anisotropy energy. To gain insight into theintrinsic unperturbed properties of these systems with-out coupling to a support or matrix, gas phase studiesof free particles are mandatory. Stern-Gerlach deflec-tion experiments7,13,20,43–46, which yield total magneticmoments with high precision20, have been the methodof choice for the determination of magnetic propertiesof free clusters for a long time. In recent experimen-tal studies, x-ray magnetic circular dichroism (XMCD)spectroscopy47–51 has been applied successfully to clus-ters that are stored in an ion trap52–55, giving access tothe intrinsic spin and orbital magnetic moments of iso-lated clusters. These studies showed a peculiar reductionof the average spin magnetic moment53 in Fe+13 as well asthe strong quenching of the orbital magnetic moment al-ready for very small iron clusters of only three atoms.Here, we show that the reduced spin magnetic moment

2

that is observed for Fe+13 is neither present in Co+13 norNi+13 clusters but is unique among the ferromagnetic 3delements in this size range. In fact, with the exceptionof Fe+13, all Fe+n , Co+n , and Ni+n clusters with n = 10− 15atoms per cluster are strong ferromagnets with magneticmoments of ≈ 1µB per empty 3d state. This particularbehavior of iron clusters might be related to the com-plex magnetic properties of bulk iron and to the existenceof antiferromagnetic or nonmagnetic phases in the bulkphase diagram56–62.

II. EXPERIMENTAL SETUP

Cluster ions are produced in a liquid nitrogen cooledmagnetron gas aggregation source, size selected in aquadrupole mass filter, and guided into a radio-frequencyquadrupole ion trap where x-ray absorption and x-raymagnetic circular dichroism (XMCD) spectroscopy47–51

of size-selected cluster ions52–55 is performed in ion yieldmode63–69at the L2,3 edges of iron, cobalt, and nickel.The total magnetic moments of size-selected gas phaseclusters are aligned by a homogeneous magnetic fieldµ0H ≤ 5 T of a superconducting solenoid that is placedaround the ion trap. The ion trap housing and electrodesare cooled down to a temperature of 4 − 6 K by a flowof liquid helium, and the cluster ions are thermalized bya constant flow of purified and pre-cooled helium buffergas in the presence of the applied magnetic field. Theion temperature T ion is typically 10 to15 K. The heliumatom density in the ion trap corresponds to 10−4 to 10−3

mbar at room temperature.Along the ion trap axis a monochromatic and ellipticallypolarized soft-x-ray synchrotron radiation beam from anundulator beamline (UE52-SGM and UE52-PGM) at theBerlin synchrotron radiation facility BESSY II is coupledin for photoexcitation of the clusters in the vicinity ofthe transition metal L2,3 edges. This interaction leadsto dipole-allowed transitions from atomic 2p core levelsinto unoccupied d and s valence states as well as to di-rect valence and core-level photoionization. In this pro-cess, 3d states are predominantly probed because of thelarge transition matrix element70,71, which leads to x-rayabsorption cross sections that are larger by one order ofmagnitude than transitions into higher ns and nd (n > 3)states. This excitation scheme allows to probe the mag-netic moments of iron, cobalt, and nickel clusters, whichare carried by the 3d electrons while magnetic contribu-tions from 4s and 4p states are negligible for clusters withmore than 10 atoms72 as well as in the bulk73–76.The 2p core hole that is created in the x-ray absorptionprocess relaxes via Auger decay cascades. This leadsto highly charged clusters, which disintegrate predomi-nantly into monomer cations63,66 that are also stored inthe ion trap. Bunches of parent ions and product ionsare extracted from the ion trap by a pulsed exit aper-ture potential, and are detected by a reflectron time-of-flight mass spectrometer with a mass resolution of

m/∆m ≈ 3000 that operates in the inhomogeneous strayfield of the superconducting solenoid and therefore ismounted in-line with the ion trap for maximum trans-mission.To obtain ion yield spectra as a measure of x-ray absorp-tion, time-of-flight mass spectra were recorded for paral-lel (σ+) and antiparallel (σ−) orientation of the magneticfield µ0H and photon helicity σ with a total data acqui-sition time of 8 − 24 s per photon energy step. Thesex-ray absorption and XMCD spectra were recorded witha target density of ≈ 5 × 107 ions cm−3, a typical pho-ton flux of 1 − 5 × 1012 photons per second, and a typ-ical photon energy resolution of 250 meV. All spectrawere normalized to the incident photon flux, monitoredwith a GaAsP photodiode mounted on-axis behind theion trap. The XMCD asymmetry was corrected for thecircular polarization degree of 90% (P3 = 0.9) of theelliptically polarized soft x-ray beam. The XMCD asym-metry was typically recorded by inversion of the photonhelicity, but also cross-checked for systematic errors byinverting the direction of the applied magnetic field, theresults of which are identical within the error bars.The incident soft x-ray beam also generates parasitic he-lium ions by direct photoionization of the helium buffergas. These helium ions where continuously removed fromthe ion trap by radio-frequency ion cyclotron resonanceexcitation of He+ in the applied magnetic field in orderto prevent the parent ions from being pushed out of thetrap by the high space charge on the ion trap axis.

III. EXPERIMENTAL METHODS

A. X-ray absorption and XMCD spectroscopy of

size-selected free cluster ions

Typical x-ray absorption spectra,

IXAS = 1/2(

I(σ+) + I(σ−))

, (1)

XMCD spectra,

IXMCD = I(σ+) − I(σ−), (2)

and the integrated 2p → 3d x-ray absorption cross sec-tions with respect to the photon energy are shown in Fig.1 for Co+13 and Ni+13 clusters. The signal-to-noise ratio inthis data is comparable to x-ray absorption spectra ofbulk metals51 even though the target density in the iontrap corresponds to only 10−6 − 10−5 atomic monolay-ers. This signal-to-noise ratio is achieved because x-rayabsorption spectra of free clusters in ion yield mode arenot obscured by substrate background absorption.For a quantitative comparison to bulk data, the contin-uum absorption of the x-ray absorption spectra shownin Fig. 1 has been scaled to the calculated cross sec-tion of the direct 2p photoionization ≈ 40 eV abovethe L3 absorption band of 1.34 Mbarn at 815 eV forcobalt, and 1.28 Mbarn at 890 eV for nickel77,78. Good

3

photon energy (eV)

760 780 800 820 840 860 880

x-ray a

bso

rptio

n / X

MC

D in

teg

ral (M

ba

rn e

V)

Co13

+ Ni13

+

0

2

4

6

-4

-2

XM

CD

(M

ba

rn)

x-ra

y a

bso

rpti

on

(M

ba

rn)

-10

-5

0

10

5

15

20

A3

C

A3 + A

2

FIG. 1. L2,3-edge x-ray absorption and XMCD spectra (solidlines) of Co+13 and Ni+13; integrated 2p → 3d x-ray absorp-tion cross sections (dashed lines); and the two step functions(dotted lines) that approximate direct 2p photoionization and2p → ns, nd (n > 3) contributions.

agreement with the absolute absorption cross section inthe corresponding bulk spectra at the L3 line of ≈ 6.5Mbarn for cobalt and ≈ 4.6 Mbarn for nickel1,51,79,80 isfound. Even though the line shape is almost bulk-like,the line width53,63,69 is narrower for the clusters investi-gated here.In addition to the x-ray absorption spectra, two stepfunctions are shown in Fig. 1. These step functions ap-proximate the intensity Ic of Rydberg (2p → ns, nd;n ≥4) and continuum (2p → ǫs, ǫd) transitions that lead tothe absorption edge. The intensity of these transitionsis subtracted from the experimental spectra to extractonly the intensity I3d of 2p → 3d transitions, as thesecontain information about the magnetic properties thatare carried mainly by the 3d electrons. As shown byHirsch et al.

71, the positioning of these two step func-tions at the peak of the L2,3 transitions, as it is donefor the bulk51, is also justified for finite ionic systems,although for the latter case the direct 2p photoionizationthreshold that marks the onset of continuum excitationsis shifted by several eV to higher photon energy.68,71,81

This approach is valid, because the x-ray absorption crosssection of transitions into higher ns, nd (n ≥ 4) states issimilar to that of direct photoionization from the 2p corelevels, but these 2p → ns, nd transitions do not shift rel-ative to the 2p → 3d excitation and therefore still createthe step edges underneath the L3 and L2 lines.71

In order to obtain quantitative magnetic moments, wellknown XMCD sum rules49–51 are used to link measuredx-ray absorption and XMCD spectra to the projectionmS of the spin magnetic moment µS

mS = −2µBnh

A3 − 2A2

C− 7

µB

~〈Tz〉 (3)

photon energy (eV)

x-ra

y a

bso

rpti

on

(M

ba

rn)

Fe10

+

690 700 710 720 730 740 750

0

2

4

6

0.30 T

0.70 T

1.20 T

1.75 T

2.50 T

3.75 T

5.00 T

applied magnetic field µ0H (T)

inte

gra

ted

XA

S

(Mb

arn

eV

)

0

10

20

30

1 2 3 4 5

FIG. 2. L2,3-edge x-ray absorption of Fe+10 at different appliedmagnetic fields. The identical line shape for 0 − 5 T showsthe absence of natural linear dichroism or spatial alignment.Integrated spectra (inset) are constant within an error of ±6%.

and to the projection mL of the orbital magnetic momentµL

mL = −4

3µBnh

A3 + A2

C(4)

onto the direction of the magnetic field µ0H at a giventemperature. Here, nh is the number of unoccupied 3dstates and 〈Tz〉 is the spin magnetic dipole term. Thequantities

A3 =

∫

L3

I3d(σ+) − I3d(σ−) dE (5)

and

A2 =

∫

L2

I3d(σ+) − I3d(σ−) dE (6)

correspond to the integrated XMCD asymmetry at theL3 and L2 absorption edges, respectively, and

C = 1/2

∫

L3+L2

I3d(σ+) + I3d(σ−) dE (7)

to the integrated 2p → 3d x-ray absorption spectrum af-ter step-edge correction. These integrals A3, A2, and Care marked in Fig. 1 with arrows.Even though 〈Tz〉 can be a significant contribution tothe spin sum rule at single crystal surfaces82, there isno contribution of 〈Tz〉 to the spin sum rule for ran-domly oriented samples83. To illustrate the situation infree transition metal clusters, Fig. 2 shows a series of x-ray absorption spectra of Fe+10 for applied magnetic fieldsranging from µ0H = 0.3 up to 5.0 T, corresponding toan alignment of the magnetic moment of 16 % to 91 %at 15 K ion temperature if a total magnetic moment of

4

bulk clusters dimers nh

Fe 3.34− 3.44 2.9− 3.5 3.25 3.3± 0.2

Co 2.5 2.3− 2.5 2.5 2.5± 0.2

Ni 1.5 0.7− 1.2 1.2 1.3± 0.2

TABLE I. Calculated number of unoccupied 3d states for bulkiron, cobalt, and nickel82,84,85; iron (n = 2− 89), cobalt (n =1− 5), and nickel (n = 2− 6) clusters74,75,86–90; and cationicdiatomic molecules91. In this work, we assume nh as given inthe last column.

36 µB is assumed. All Fe+10 x-ray absorption spectra inthis series exhibit an identical line shape, independentof the applied magnetic field. Small differences in thepeak intensities at the L2,3 edges are caused by statis-tical variations but are not correlated to the magneticfield strength. The approximate absolute x-ray absorp-tion cross sections that were obtained by scaling thesespectra to the calculated 2p direct photoionization crosssection of iron, 1.45 Mbarn77,78 at 748 eV, are in goodagreement with the absolute cross section of ≈ 7 Mbarnat the L3 line of bulk iron spectra1,51,80. The integratedx-ray absorption spectrum of Fe+10 is constant within 6 %,as can be seen in the inset of Fig. 2. If 〈Tz〉 were large, i.e.,if bonding and charge distribution in the clusters werestrongly anisotropic, then clusters that would be spa-tially aligned by the magnetic field should show a stronglinear dichroism in their x-ray absorption spectra. Thisis not the case as can be seen from Fig. 2. Hence, theclusters are randomly oriented in the ion trap and 〈Tz〉averages to zero.

B. Constant number nh of unoccupied 3d states in

small clusters

The number nh of empty 3d states is a parameter inthe XMCD sum rules49–51 (Eqs. 3 and 4) that is re-quired to obtain the magnetic moment per atom. Forthe cluster ions investigated here, nh for each elementhas been taken from the available theoretical values forthe number of unoccupied 3d states in iron, cobalt,and nickel bulk metals73,82,84,85,92–94, clusters74,75,86–89,and diatomic molecular cations91. Calculations for ironclusters88 have shown that nh is close to the bulk valueand does not undergo significant variations over a largerange of cluster sizes. This also applies to calculated val-ues of nh for cobalt and nickel clusters, which are againclose to the corresponding bulk values as can be seenfrom Table I. We thus assume nh = 3.3 ± 0.2 for iron,nh = 2.5 ± 0.2 for cobalt, and nh = 1.3 ± 0.2 for nickelclusters.Experimentally, nh can be determined from the chargesum rule1,70,95–98 of x-ray absorption

C ∝ nh (8)

cluster size (atoms per cluster)

151413121110

Fen

+

Con

+

Nin

+

nu

mb

er

of

3d

ho

les

3

2

1

4

0

FIG. 3. Experimentally determined variation in the numbernh of unoccupied 3d states for Fe+n , Co

+n , and Ni+n clusters.

Within the error bars, nh is constant for clusters of a givenelement.

that relates nh to the integrated x-ray absorption crosssection C (cf. Eq. 7) of 2p → 3d transitions. This pro-portionality can also be seen in Fig. 1, where the inte-grated x-ray absorption cross section C of 2p → 3d tran-sitions is 17.7 Mbarn eV for Co+13, but 8.4 Mbarn eV forNi+13, reflecting the larger number of unoccupied 3d statesnh = 2.5 for cobalt than nh = 1.3 for nickel clusters.For Fe+n , Co+n , and Ni+n with n = 10 − 15, the experi-mentally determined variation in the number of empty3d states is shown in Fig. 3. The values given here weredetermined from the charge sum rule1,70,95–97, cf. Eq. 8,by integrating the x-ray absorption spectrum, and cal-ibrating the average integrated intensity C of 2p → 3dtransitions (cf. Eq. 7) with the nh value75,82,84–87,89,91

from Table I for a given element. The scatter in the ex-perimentally determined nh for different cluster sizes ofthe same element in Fig. 3 is of the same order of magni-tude as the scatter for an individual cluster at differentmagnetic fields as shown in the inset of Fig. 2 for thecase of Fe+10. This implies that the observed scatter inthe number of unoccupied 3d states is due to experimen-tal uncertainties rather than to a real variation of nh,and the experimental data thus indeed reveal a nearlyconstant nh within the error bars for a given element inthe cluster size range considered here.The experimental proportionality constant of 6.4 − 7.3Mbarn eV per empty 3d state is similar for iron, cobalt,and nickel clusters, but is only ≈ 2/3 of the reported bulkvalue1 in our data. This might be related to the partialion yield detection technique, which is a good approxi-mation of, but not identical to, x-ray absorption. Since2p → 3d transitions and direct 2p photoionization lead tocharge states of the core-excited cluster that differ by 1,the product ion distribution after multiple Auger decayand fragmentation will not be identical for both excita-tion channels. Direct 2p photoionization typically leadsto product ions with a larger charge-to-mass ratio q/mthan 2p → 3d transitions. Since all spectra shown herewere recorded on the dominant singly charged metal ion,

5

photon energy (eV)

Co10

+

XM

CD

(M

ba

rn)

x-ra

y a

bso

rpti

on

(M

ba

rn)

µ0H

0

2

4

6

-4

-2

XAS and XMCD full scan

XAS and XMCD sampling

0.5 T

5.0 T

760 780 800 820770 810790750

FIG. 4. X-ray absorption and XMCD spectra taken atµ0H = 5 T (solid lines) and low resolution peak intensityscans (triangles) of the XMCD and x-ray absorption signalat µ0H = 0.5 − 5.0 T, from which magnetization curves arederived.

the continuum step edge might be slightly overestimatedin the x-ray absorption spectra shown here. When scal-ing the direct 2p photoionization to calculated subshellphotoionization cross sections77,78, the absolute absorp-tion cross sections of 2p → 3d transitions, given in Figs.1 and 2, would thus be underestimated. However, thepartial ion yield detection scheme only changes the ratioof 2p → 3d transitions, I3d(σ+) + I3d(σ−) to direct pho-toionization, Ic, but does not alter the relative intensitiesof I3d(σ+) and I3d(σ−) which enter into the XMCD sumrules via Eqs. 5 to 7 after the step edge Ic has been sub-tracted. Hence, the magnetic moments derived from theXMCD sum rules are not affected by partial ion yielddetection.

C. Magnetization curves, magnetic moments, and

electronic temperature

It is important to recall that XMCD spectroscopy, justlike Stern-Gerlach experiments, does not determine ab-solute magnetic moments µ of size selected clusters, butmeasures the magnetization m as a response to an ap-plied magnetic field at a given temperature. The mag-netic moment is then deduced from this magnetization.For a precise determination of magnetic moments andtemperature of the clusters, magnetization curves havebeen measured for Fe+13, Co+10−15, and Ni+13 by recordingthe magnetization as a function of the applied magneticfield µ0H at constant temperature, where the magneticfield, applied along the z direction, acts on the totalmagnetic moment µJ = g µB/~ Jz of the cluster, with〈Jz〉 = 〈Lz〉 + 〈Sz〉 in LS coupling, where 〈Lz〉 and 〈Sz〉are derived from XMCD sum rules for a given appliedmagnetic field.

applied magnetic field µ0H (T)

ma

gn

eti

zati

on

pe

r a

tom

(µ

B)

Co10

+

0 5321 4

0.0

0.5

1.0

1.5

2.0

2.5

0.0

0.1

0.2

0.3

0.4

mL /

mS

mJ

mL

mS

mL

/ mS = 0.25 ± 0.01

FIG. 5. Upper panel: Total (mJ , filled circles), spin (mS, lightgray circles), and orbital (mL, open circles) magnetization ofCo+10 as a function of the applied magnetic field. A Brillouinfit to the total magnetization mJ (solid line) yields the clusterion temperature T ion = 14.6 ± 0.3 K and the total magneticmoment µJ = 3.15 ± 0.09 µB per atom. The same Brillouinfit is scaled to mS and mL as a guide to the eye.Lower panel: The average ratio of mL/mS = µL/µS = 0.25±0.01 (filled triangles and dashed line) is constant within theerror bars and is used to decompose µJ into µS and µL. Theerror bars increase for smaller applied magnetic fields becauseof decreasing magnetization.

In practice, magnetization curves have been derived bysampling and reconstructing the full XMCD and x-rayabsorption spectra at the L2,3 peak intensities with lowphoton energy resolution of 1250 meV, as illustrated inFig. 4. Since natural dichroism has not been observed inthese spectra at different magnetic fields, i.e., the lineshape does not change as shown in Fig. 2, the integratedXMCD and x-ray absorption cross sections (A3, A2, andC in Fig. 1) are proportional to the peak intensities of theL2,3-edges of the XMCD (open triangles in Fig. 4) andx-ray absorption (solid triangles in Fig. 4) curves thatare repeatedly sampled at four different photon energiesfor 60 s each. As expected, the XMCD signal diminisheswith lower magnetic fields while the x-ray absorption in-tensities remain constant, confirming stable and repro-ducible experimental conditions along the acquisition ofa complete magnetization curve. The sum and differenceof the sampled L3 and L2 peak intensities at µ0H = 5T were scaled to the corresponding integrals A3, A2, andC of full spectra measured at µ0H = 5 T, shown as asolid line in Fig. 4, for the application of XMCD sumrules. The same scaling factors were then used to con-vert peak intensities into integrals A′

1, A′

2, and C′ formagnetic fields µ0H < 5 T.As an example, the magnetization curve mJ (µ0H) ofCo+10 is shown in Fig. 5 in the 0 to 5 T range. A fit to the

6

experimental data with a Brillouin function, shown as asolid line in the upper panel of Fig. 5, yields the electroniccluster ion temperature T ion and the total magnetic mo-ment µJ per atom as independent fit parameters. Thetotal magnetization mJ in Fig. 5 is also decomposed intospin and orbital contributions mS and mJ for illustration,although µ0H only acts on J but neither on L nor on Sseparately. The corresponding spin and orbital magneticmoments µS and µL per atom are derived from a decom-position of the total magnetic moment mJ according tothe mean value of the experimentally determined ratiomL/mS of orbital to spin magnetization that is recordedfor a given cluster at different applied magnetic fields, asis shown for Co+10 in the lower panel of Fig. 5.The analysis of the magnetization curves does not onlyyield the magnetic moments µJ , µS , and µL but also theelectronic temperature T ion of the cluster ions. To thebest of our knowledge, this is the only method so far todetermine the electronic temperature of isolated systemsin an ion trap53. The typical electronic cluster tempera-ture under our experimental conditions is T ion = 12 ± 3K, and is increased by 7±3 K over the ion trap electrodetemperature Ttrap = 5 ± 1 K because of radio frequencyheating in the presence of a helium buffer gas99. Theelectronic temperature corresponds to the vibrational orrotational cluster ion temperature100–104 because thesedegrees of freedom are in thermal equilibrium by multi-ple collisions with the helium buffer gas.

IV. RESULTS

A. Spin and orbital magnetic moments

In Fig. 6, the total, spin, and orbital magnetic mo-ments are presented for iron53, cobalt, and nickel clustersin the size range of n = 10−15. For cobalt, magnetic mo-ments have been determined from magnetization curvesfor all cluster sizes, which leads to the smallest error. Ex-cept for Fe+10 and Ni+13, the data for iron and nickel clus-ters were taken at a fixed ion trap temperature and fixedmagnetic field of 5 T. To obtain magnetic moments, thetotal magnetization that was obtained from the XMCDanalysis under these conditions was Brillouin corrected.The error bars in Fig. 6 correspond to statistical intensityvariations of the XMCD spectra or, as for cobalt clusters,from Brillouin fits to magnetization curves. Systematicerrors, e.g. uncertainties in the number of unoccupied 3dstates (∆µJ ≈ 0.2 µB) and in the position of the two stepfunctions (∆µJ ≈ 0.1 − 0.2 µB) sum up to 0.3 µB peratom for cobalt and 0.4 µB per atom for iron and nickelclusters but are omitted from Fig. 6.Table II lists the average experimental 3d spin and or-bital magnetic moments per atom and per cluster. Eventhough XMCD spectroscopy at the L2,3 edges of 3d tran-sition elements is only sensitive to the 3d magnetic mo-ments, these values will be close to the total magneticmoments because 4s and 4p electron contributions to the

cluster size (atoms per cluster)

ma

gn

eti

c m

om

en

t p

er

ato

m (µ

B)

ma

gn

etic m

om

en

t pe

r 3d

ho

le (µ

B )

10 12 1411 1513

Nin

+

µJ

µS

µL

0.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

2.0

Con

+

µJ

µS

µL

0.0

0.5

1.0

0.0

1.0

2.0

3.0

1.5

Fen

+

µJ

µS

µL0.0

1.0

2.0

3.0

4.0

0.0

0.5

1.0

1.55.0

FIG. 6. Total (µJ , open symbols), spin (µS, filled symbols),and orbital (µL, open symbols with dot) magnetic momentsper atom and per unoccupied 3d state of iron, cobalt, andnickel clusters. With the exception of Fe+13, all clusters carry≈ 1 µB per 3d hole.

spin magnetic moment are expected to be negligible72–76.

Within the range defined by the error bars of these exper-imental magnetic moments, only integral values for thespin multiplicity 2S and the orbital angular momentumL per cluster are allowed because the angular momen-tum that leads to the magnetic moment is quantized. Afurther constraint on the spin magnetic moment is im-posed by the total number of valence electrons ne percluster, which defines whether the spin multiplicity ofteh cluster is odd or even. For Fe+n , Co+n , and Ni+n with3d6 4s2, 3d7 4s2, and 3d8 4s2 atomic configurations, thetotal number of electrons per cluster is listed in Table IIas ne = 8n− 1, ne = 9n− 1, and ne = 10n− 1, respec-tively. The values of 2S and L of iron, cobalt, and nickelclusters that are compatible with these constraints arealso listed in Table II. Note that for the determination ofµS and µL, the number of empty 3d states, given in TableI, is assumed to be constant, even though nh will haveto vary slightly with cluster size under the constraints ofan integral number of atoms per cluster n and integertotal spin multiplicity 2S for a strong ferromagnet with-

7

ne µS µL µclS µcl

L 2S L 2S theo L theo

(µB) (µB) (µB) (µB) (~) (~) (~) (~)

Co+10 89 2.54 ± 0.09 0.61 ± 0.03 25.4± 0.9 6.1± 0.3 25 6

Co+11 98 2.41 ± 0.14 0.69 ± 0.04 26.5± 1.5 7.6± 0.5 26; 28 8

Co+12 107 2.68 ± 0.19 0.67 ± 0.05 32.1± 2.3 8.0± 0.6 31; 33 8 21 a

Co+13 116 2.49 ± 0.14 0.60 ± 0.03 32.4± 1.8 7.8± 0.4 32; 34 8 26 a

Co+14 125 2.58 ± 0.19 0.53 ± 0.04 36.2± 2.6 7.4± 0.6 35; 37 7; 8

Co+15 134 2.49 ± 0.19 0.38 ± 0.04 37.3± 2.8 5.7± 0.5 36; 38; 40 6

Fe+10 79 3.19 ± 0.19 0.26 ± 0.04 31.9± 1.9 2.6± 0.4 31; 33 3 29 a b 1.4 − 1.8 b

Ni+13 129 1.20 ± 0.14 0.32 ± 0.06 15.6± 1.9 4.2± 0.8 15; 17 4; 5 9 a 3.4 − 4.2 c

a from Ref. 105.b from Ref. 76.c from Ref. 106.

TABLE II. Total number ne of valence electrons per cluster; experimentally determined spin and orbital magnetic momentsper atom, µS and µL, and per cluster, µcl

S and µclL ; spin multiplicities 2S and orbital angular momenta L per cluster that are

compatible with the experimental error bars, compared to available theoretical values 2S theo and L theo of the total spin76,105

and orbital76,106 angular momentum. Potential contributions of 4s and 4p electrons to the experimental spin magnetic momentmS are neglected. Values of 2S ≈ n · nh are marked in bold face.

out sp contributions to the total magnetic moment. Thisvariation is below the error bars of the experimentallydetermined nh data that is shown in Fig. 3. The valuesof 2S ≈ n · nh, which are close to a pattern where eachatom contributes on average 1/2nh ~ to the total spin,are marked in bold face in Table II.From these numbers, the exact spin state can be givenexperimentally for Co+10 as 2S = 25, while two differ-ent spin states are possible within the experimental errorbars for Co+11 to Co+14; Fe+10, and Ni+13. For Co+15, threedifferent spin states are compatible with the experimen-tal data. As can be seen from the available theoreticalvalues, also included in Table II, the spin magnetic mo-ments seem to be underestimated by DFT in comparisonto the experimental data for iron, cobalt, and nickel, witha difference of 0.4−0.8 µB per atom. In this respect, theexperimental data presented here can serve as a bench-mark for further development of theoretical methods topredict spin and orbital magnetic moments.

B. Magnetic anisotropy energy of transition metal

clusters

The magnetic anisotropy energy EMAE is a conse-quence of spin–orbit and orbit–lattice coupling, and isresponsible for the preferred alignment of the magneticmoment along easy crystallographic axes. It is thereforean important property of magnetic materials, in partic-ular in view of technological applications. For transi-tion metal dimers, large magnetic anisotropy energies inthe 1 − 10 meV range have been predicted107,108, whiletransition metal clusters of ≈ 1000 atoms were foundexperimentally109 to have magnetic anisotropy energiesthat are comparable to or even lower than the bulk val-

ues. Even though we do not determine EMAE directly,we can give an estimate of the upper limit of EMAE

in small iron, cobalt, and nickel clusters from energyconsiderations53 and from the absence of natural lineardichroism in the x-ray absorption spectra taken at dif-ferent magnetic fields µ0H , as presented in Fig. 2 forthe case of Fe+10. The absence of linear dichroism is dueto a random orientation of the clusters inside the iontrap. This implies that the magnetic anisotropy energyEMAE is smaller than the thermal (rotational) energyErot = 1/2 kBT per degree of freedom53. Under theseconditions, the total magnetic moment of the clustercan overcome the anisotropy barrier to rotate freely andalign with the applied magnetic field, independent of thespatial cluster orientation. This leads to the observedsuperparamagnetic14,15 behavior.With the knowledge of the cluster ion temperature frommagnetization curve thermometry, we can deduce an up-per limit of the magnetic anisotropy energy,

EMAE < 1/2 kBT ion. (9)

Since this upper limit on EMAE is mainly determined bythe cluster ion temperature, EMAE is most likely overes-timated. Therefore, Eq. 9 yields EMAE ≪ 52 µeV peratom for a fixed rotation axis of Co+10 and Ni+10 clusters.For Fe+10, a slightly increased upper limit of EMAE ≪ 65µeV per atom is estimated53 from Eq. 9 because of differ-ent experimental conditions with higher cluster ion tem-perature. These experimental upper limits on EMAE areclose to the magnetic anisotropy energy of 60 µeV forhexagonal close-packed cobalt110,111 with a strongly pre-ferred c axis, which leads to an anisotropy energy thatis much larger than EMAE = 1.4 µeV in body-centeredcubic iron and EMAE = 2.8 µeV in face-centered cubicnickel110,111. For nearly spherical and compact clusters

8

in the size range considered here, we expect EMAE to besmaller than in crystalline bulk because of the absenceof strongly preferred axes. For the same reason, we alsoexpect magnetic anisotropy energies in iron, cobalt, andnickel clusters to be closer to each other than in bulk,because the difference in EMAE should mainly be due tothe amount of orbital magnetism. This would lead toEMAE (Co) > EMAE (Ni) ≥ EMAE (Fe) from the datapresented in Table II. DFT calculations by Alvarado-Leyva et al. for the cationic Fe+13 cluster obtain a mag-netic anisotropy energy of EMAE = 66 µeV per clusteror 5 µeV per atom74, which is close to the bulk valueand consistent with our interpretation of the experimen-tal data.In contrast to deposited clusters or adatoms, where themagnetic anisotropy energy is in the meV range21,112

and is strongly enhanced by the two-dimensional geom-etry or the interaction with the support that induceslarge orbital magnetic moments, the intrinsic magneticanisotropy energy is about two to three orders of magni-tude lower in free clusters.

V. DISCUSSION

A. Strong ferromagnetism in Fe+n , Co+n , and Ni+n

clusters

Fig. 6 displays the measured total, spin, and orbitalmagnetic moments per atom and per unoccupied 3dstate. As can be seen, all clusters except for Fe+13 exhibita completely filled 3d majority spin band that, within theerror bars, leads to a spin magnetic moment of µS = 1µB per unoccupied 3d state, marked by the dashed linesin Fig. 6. These maximized spin magnetic moments re-veal a strong ferromagnetism in iron, cobalt, and nickelclusters, which is in contrast to the bulk metals. For com-parison, the majority spin bands of bulk iron, cobalt, andnickel are not completely filled but the bulk spin mag-netic moments correspond to approximately 0.62, 0.63,and 0.38 µB per 3d hole1,73,82, respectively. Interest-ingly, the total number of empty 3d states (cf. Table Iand Fig. 3) is predicted88 to be nearly identical in smallclusters and in the bulk, which indicates a similar amountof spd hybridization, but only the distribution of 3d holesover the majority and minority states is different. Theenhancement of the spin magnetic moments of clustersover the bulk values would thus indeed be caused by aredistribution of electrons from minority into majority 3dstates because of a narrower 3d band as a consequence ofthe lower average coordination in clusters13,113,114, wheremost of the atoms reside at the surface in the size rangeconsidered here.

B. Reduced spin magnetic moment of Fe+13

Fe+13 is the only exception from this general finding ofstrong ferromagnetism. As reported previously53, Fe+13has an average spin magnetic moment of µS = 2.4 ± 0.4µB per atom, much closer to the bulk value than anyother cluster in this size range. Since the number nh

of 3d holes is nearly constant for Fe+n (n = 10 − 15),as has been shown experimentally in Fig. 3, the aver-age spin magnetic moment is only 0.73 ± 0.12 µB per3d hole in Fe+13. This value is significantly reduced by∆µS = 1.0± 0.5 µB per atom in comparison to adjacentcluster sizes. In contrast to iron, the situation is differentfor cobalt and nickel clusters. Here, our data give spinmagnetic moments of Co+13 and Ni+13 that do not show asignificant reduction when compared to neighboring clus-ter sizes.There is a controversy about the origin of the reducedspin magnetic moment of Fe+13 in the literature. DFTmodels of cationic Fe+13 find either an antiferromag-netically coupled spin magnetic moment of the centralatom74,76 or a ferromagnetic alignment of the 3d spinsat all atoms but a nearly quenched spin magnetic mo-ment of the central atom75,90. In spite of these differ-ences, all these studies predict the same average magneticmoment of µS = 2.69 µB per atom74–76,90 for Fe+13 andreproduce our experimental value of µS = 2.4 ± 0.4 µB

within the error bars, cf. Fig. 7, even though the mod-eled reduction of ∆µS = 0.4 µB per atom for Fe+13 incomparison to Fe+12 and Fe+14 is less pronounced in theDFT calculations75,76,90 than in the experiment. Notonly for Fe+13 but also for other Fe+n clusters, all exist-ing calculations74–76,90 determine identical spin magneticmoments as can be seen from Fig. 7, although the ge-ometric and electronic ground state structures of Wuet al.

75 and Gutsev et al.90 differ from those found by

Alvarado-Leyva et al.74 and Yuan et al.

76 These distinc-tions in the calculated geometric and electronic struc-tures lead to different explanations about the origin of thereduced spin moment of Fe+13: While Wu et al. concludea symmetry-driven spin quenching in a highly-symmetricicosahedral structure, Alvarado-Leyva et al. report a Th

distorted icosahedral structure of Fe+13 in combinationwith an electronic shell closure that leads to antiferro-magnetic spin coupling74.It should be noted that the results of our XMCDstudy yield average spin and orbital magnetic momentsbut do not give direct access to the relative orienta-tion of spins. Therefore, both possible configurations,i.e., antiferromagnetic74,76 or ferromagnetic with nearlyquenched spin of the central atom75,90, are compatiblewith our experimental results within the error bars ofthe spin magnetic moment. Indirect evidence for anti-ferromagnetic spin coupling in Fe+13 might be obtainedfrom the reduced number nh = 2.92 of unoccupied 3dstates that is predicted for Fe+13 in the studies of Wuet al. and Gutsev et al. in contrast to nh = 3.34 predictedby Alvarado-Leyva et al. in their study. Even though a

9

slight reduction of nh in Fe+13 cannot be excluded withinthe experimental error bars given in Fig. 3, we do not seeexperimental evidence for a strong deviation of Fe+13 fromthe constant number nh ≈ 3.3 of unoccupied 3d states iniron clusters89.Also for neutral iron clusters, DFT modeling115,116 insome cases has assigned a reduction in the average mag-netic moment of neutral Fe13 to an antiferromagnetic spincoupling of the central atom to the shell atoms, and hasdemonstrated that the antiferromagnetic state becomesmore stable as the interatomic distance decreases116,117.Interestingly, more recent calculations75,76,90,118 ratherfind a larger spin magnetic moment in neutral Fe13 thanin Fe12 or Fe14 by 0.23 − 0.24 µB per atom, which isincreased because of larger 4s contributions to the mag-netic moment75,76,90 in the calculations.Similar to the antiferromagnetic alignment of the cen-tral atom in several studies115,116 of neutral Fe13,the reduction of the average spin magnetic momentin cationic Fe+13 is correlated with a compression ofthe interatomic distances by 2 − 3 % when comparedto the modeled ferromagnetic ground state of neutralFe13 as well as to cationic Fe+12 and Fe+14 in DFTmodeling74–76,90,116,117,119. This reduction in bond lengthcorresponds to a spherical volume compression of V/V0 =0.91 − 0.94 and might be correlated to the magneticproperties via the bulk phase diagram of iron, wherea transition from the ferromagnetic body-centered cu-bic (bcc) α phase to the nonmagnetic hexagonal closepacked (hcp) ǫ phase of iron occurs around 12 − 16GPa at room temperature56–61. This transition occursaround a volume compression of V/V0 = 0.94120–122,which corresponds to the calculated compression ratioin Fe+13

74–76,90,116,117,119, even though the modeled Th

symmetry of Fe+1374–76,90can be interpreted as a distorted

fcc74 rather than hcp structure. The face-centered cubic(fcc) γ phase of bulk iron is stable only above the Curietemperature of α iron and is paramagnetic, but an anti-ferromagnetic fcc phase123 of iron with a Neel tempera-ture of 67 K can be stabilized in iron precipitates124–126.Complex antiferromagnetic order in thin films of fcciron grown on copper (001) has also been observedexperimentally127–130.In bulk cobalt, a comparable transition from the ferro-magnetic hcp ǫ phase to the nonmagnetic fcc β phase oc-curs at much higher pressure of 80− 130 GPa and highervolume compression of V/V0 ≈ 0.7562,121,131–136. Whenapproximating Co+13 as a sphere, this volume compressionwould correspond to a large radial bond compression of≈ 10% that is not expected to occur in Co+13 and might bea hint at why no reduction of the spin magnetic momentis observed in this case. Similarly, no phase transitionhas been observed in bulk nickel up to a pressure of 200GPa and a volume compression of V/V0 ≈ 0.6562,121,137,where ferromagnetic fcc nickel is still stable. Therefore,neither a change of magnetic order nor a reduction of themagnetic moment of the central atom by bond compres-sion is likely to occur in Ni+13. This way, the bulk phase

Con

+

cluster size (atoms per cluster)

spin

ma

gn

eti

c m

om

en

t p

er

ato

m (µ

B)

0.0

1.0

2.0

3.0

18 206 10 148 1612

Fen

+

0.0

1.0

2.0

3.0

4.0

exp. data (this work, +)

Yuan et al. (+)

Alvarado-Leyva et al. (+)

Gutsev et al. (+)

Datta et al.

Rodríguez-López et al.

Ma et al.

Guevara et al.

Fan et al.

Gutsev et al. (+)

exp. data (this work, +)

Nin

+

0.0

1.0

2.0exp. data (this work, +)

Reddy et al.

Andriotis et al.

Aguilera-Granja et al.

Gutsev et al. (+)

FIG. 7. Calculated spin magnetic moments (open symbols)of iron, cobalt, and nickel clusters72,75,105,115,119,138–144 com-pared to spin magnetic moments, derived in this XMCD study(filled symbols). The + symbol denotes results for cationicclusters. Systematic errors are not shown.

diagrams elucidate why the peculiar behavior of antifer-romagnetic alignment or spin quenching at the centralatom of the thirteen-atom cluster is observed only foriron but does not show up in cobalt or nickel clusters,although the bulk phase diagrams do not explain whatmechanism would be responsible for such a spontaneousbond contraction and change of magnetic order.Even for iron, this unexpected magnetic behavior isonly observed for Fe+13. This finding might either be re-lated to electronic shell closure in the spin-up and spin-down states, or to the geometric structure with onecentral atom and 12 shell atoms. The modeled elec-tronic structure and geometry of Fe+13 should, however,be taken with caution: Even though many likely geomet-ric structures have been explored in electronic structurecalculations74–76,90, no unbiased global geometry opti-mization has been performed for cationic iron clusters,to the best of our knowledge. Apart from adsorption ortitration studies145–147 that cannot give detailed informa-tion on interatomic distances, no experimental structure

10

determination has been reported for free iron, cobalt, andnickel clusters in this size range so far. The experimen-tal bond energies of cationic iron clusters show a localmaximum148 at Fe+13, which is in agreement with its en-hanced abundance in mass spectrometry149 studies. Theionization potentials of neutral iron clusters also show apronounced maximum around Fe13 and Fe14

150–152. Totest the calculated bond length compression in Fe+13 ex-perimentally, trapped ion electron diffraction153, infraredmultiphoton dissociation154, or extended x-ray absorp-tion fine structure studies155 would be required with theirsensitivity to geometric structure and interatomic dis-tances.

C. Comparison of experimental and theoretical

spin magnetic moments of Fe+n , Co+n , and Ni+n

clusters

For iron, cobalt, and nickel clusters, nearly all theoret-ical studies performed on neutral72,75,115,119,138–144,156 aswell as on cationic74–76,90,105 clusters seem to underesti-mate the average spin magnetic moments by 0.4−0.8 µB

per atom, as can be seen in Fig. 7 and Table II. Such anunderestimation may originate from the known inabilityof standard approximations to the exchange-correlationfunctional to describe highly correlated electrons as inthe 3d transition metals. This problem is avoided instudies of Rodrıguez-Lopez et al. and Aguilera-Granjaet al.

72,138 by using semi-empirical many-body potentialsand a self-consistent tight-binding method, where elasticconstants or hopping and exchange integrals are fitted toreproduce bulk properties, at the expense of an ab ini-

tio approach. As can be seen, the calculated magneticmoments of Rodrıguez-Lopez et al. and Aguilera-Granjaet al. (open squares in Fig. 6) agree better with the ex-perimental magnetic moments that are determined in ourstudy. However, the results of semi-empirical calculationsdepend strongly on the parametrization and might not beable to reproduce the significant change in the geomet-ric and electronic structure that is apparently inducedby a single elementary charge difference in neutral andcationic Fe+13 as found by ab initio calculations74–76,90,105.

D. Reduction of atomic spin and orbital magnetic

moments in small iron, cobalt, and nickel clusters

Fig. 8 illustrates the reduction of spin magnetic mo-ments and the quenching of orbital magnetic momentsfor iron, cobalt, and nickel clusters in comparison to thefree atom values. Here, the experimentally determinedspin and orbital magnetic moments have been normal-ized to the corresponding 3d spin magnetic momentsof the atomic 3d6, 3d7, and 3d8 configuration of iron,cobalt, and nickel. Normalization to the atomic insteadof the cationic configuration was chosen, even though forCo/Co+ and Ni/Ni+ the configuration changes from 3d7

cluster size (atoms per cluster)

18161412108 bulk

no

rma

lize

d m

ag

ne

tic

mo

me

nt

(µB)

0.6

0.4

0.2

0.8

0.0

Fen

+

Con

+

Nin

+

1.0

83 %

65 %

FIG. 8. Spin (open symbols) and orbital magnetic moments(filled symbols) scaled to atomic values of the Fe (3d6), Co(3d7), and Ni (3d8) configurations. Dashed lines (Fe, Co:83%; Ni: 65%) mark the ratio of unoccupied 3d states inatoms and clusters.

to 3d8 and from 3d8 to 3d9 (Ref. 71, 157–160) but forclusters larger than n = 10 atoms with ≥ 60 3d valenceelectrons, the effect of a single s−d promotion on the to-tal number of 3d electrons is below 2 % and the error thatis introduced by this normalization can be neglected. ForFe/Fe+, the 3d6 configuration is identical for atoms andcations71,161,162.As can be seen in Fig. 8, the spin magnetic moments ofiron, cobalt, and nickel clusters are reduced to a simi-lar amount of 60 − 90 % of their atomic spin magneticmoments of 4, 3, and 2 µB per atom, respectively. Sinceclusters in this size regime as well as atoms carry the max-imum spin magnetic moment of µS = 1.0 µB per 3d holebecause of filled majority bands for clusters or Hund’srules for atoms, the observed reduction in the spin mag-netic moment is simply caused by a different number nh

of unoccupied 3d states. Therefore, the average reductionof the spin magnetic moments that is shown in Fig. 8 isclose to the ratio of 3d holes in atoms and clusters, whichis 0.83 for iron, 0.83 for cobalt, and 0.65 for nickel86–89. Inthe bulk metals, spin magnetic moments of iron, cobalt,and nickel are further reduced by the formation of wider3d bands, which creates empty states in the 3d majorityspin bands with average spin magnetic moments µS < 1µB per unoccupied 3d state. Even though nh is alreadybulk-like in small clusters, the distribution of unoccupiedstates over the majority and minority states is still differ-ent from the bulk but changes with cluster size. Conse-quently, the magnetic moments of neutral clusters reachthe bulk value when the formation of bulk-like spd bandsleads to the loss of strong ferromagnetism around 400 to600 atoms per cluster9,11,13,14,45.In contrast to the spin moment, orbital magnetic mo-ments of bulk iron, cobalt, and nickel are stronglyquenched by the crystal field and are only restored by thespin–orbit interaction to 2-3 % of their atomic values163.This quenching is already very pronounced in smallclusters53 with n ≥ 3 because of the reduced rotational

11

cluster size (atoms per cluster)

ma

gn

eti

c m

om

en

t p

er

ato

m (µ

B)

mL /

mS

8 12 1610 14

Con

+

Con

+

µJ

µS

Λl

0.0

0.1

0.2

0.3

0.4

0.5this work

Peredkov et al.

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0

FIG. 9. Upper panel: Comparison of spin, orbital, and to-tal magnetic moments of the present (black symbols) and aformer (light gray symbols, Ref. 52 ) XMCD study on freecobalt clusters. Lower panel: Good agreement of the ratio ofspin-to-orbital magnetization of the present study and that ofRef. 52.

symmetry. Different from bulk bcc iron, hcp cobalt andfcc nickel, the modeled structures of iron, cobalt, andnickel clusters are quite similar90,115,139,140,164. Since thestrength of the crystal field is nearly constant along the3d transition metal series163, the orbital magnetic mo-ments of iron, cobalt, and nickel clusters are reducedto a similar amount of 5 − 25 % of their atomic value.Even though the orbital magnetic moments are stronglyquenched when compared to the atom, where µL = 2 µB

for iron and µL = 3 µB, for cobalt and nickel, they aresignificantly enhanced when compared to the bulk val-ues. For clusters, we generally find µL(Co) > µL(Ni) ≥µL(Fe).The enhancement of the orbital magnetic moments inclusters compared to bulk is most likely due to a reducedcrystal field interaction that is caused by a lower coordi-nation of cluster surface atoms. In a similar way, surfaceenhanced orbital magnetic moments of µL = 0.06 − 0.3µB have been calculated for iron, cobalt, and nickel sur-faces and clusters76,106,113.

E. Coupling of spin and orbital angular momenta

In a previous XMCD study of size-selected cobalt clus-ter ions by Peredkov et al.

52, decoupling of the spin andorbital angular momenta has been postulated, becausethe authors found different Langevin scaling for orbital

and spin magnetic moments in their experimental data52.Decoupled spin and orbital angular momenta would besurprising as these would not only require significant in-teratomic orbit–orbit coupling for a total µL to inter-act with the magnetic field, but this decoupling wouldalso imply that the interaction of the spin magnetic mo-ment with the applied magnetic field were much strongerthan the 3d spin–orbit interaction in the 3d transition el-ements. However, the energy gain by alignment of a 2.5µB spin magnetic moment in a field of 7 T is of the orderof 1 meV, which is small compared to the 3d spin–orbitcoupling energy of 50− 100 meV per atom165,166 in iron,cobalt, and nickel.Experimentally, the expected coupling of spin and orbitalangular momenta, S and L, to a total angular momen-tum Jcan be shown from the field dependence of theratio of orbital to spin magnetization. This mL/mS ra-tio is shown in the lower panel of Fig. 5 for Co+10 ver-sus the applied 0 − 5 T magnetic field. As can be seen,mL/mS = 0.25±0.01 is constant and independent of theapplied magnetic field. This finding is the same for allclusters investigated and confirms the well-known Russel-Saunders coupling but contradicts the unexpected find-ing of Peredkov et al.

After rescaling for LS coupling, the spin and orbital mag-netic moments of Co+n in the study of Peredkov et al.

agree with the data presented here within the error bars,even though the magnetic moments of Peredkov et al.

seem to be systematically lower than in our study, asshown in the upper panel of Fig. 9. Most important, theratio mL/mS of orbital to spin magnetization of Co+nin the lower panel of Fig. 9 agrees very well, as thisratio can be determined from the XMCD sum rules toa higher precision than spin and orbital magnetic mo-ments separately because of the cancellation of potentialerrors in the degree of circular polarization, in the num-ber of unoccupied 3d states, or in the normalization tothe isotropic x-ray absorption spectrum. This quantita-tive agreement of two independent studies confirms thereliability of XMCD studies of size-selected cluster ions.

F. Comparison of XMCD results to Stern-Gerlach

experiments

In a size range that is similar to the one consideredhere, Stern-Gerlach deflection experiments yield totalmagnetic moments of µJ ≈ 3.0 − 5.5 µB for neutraliron clusters of 10 to 50 atoms7,9,11,14,20,43,167; µJ ≈2.25 − 3.9 µB for neutral cobalt clusters of 10 to 50atoms8,11,14,20,44,45,139,168–170; and µJ ≈ 0.8 − 1.3 µB

for neutral nickel clusters of 10 to 15 atoms10,11,13,14,171.Even though there is a large scatter in the variousStern-Gerlach data, our average XMCD results for iron(µJ ≈ 3.5 µB) and cobalt (µJ ≈ 3.0 µB) cluster ions fallwell within the range spanned for neutral clusters. Fornickel clusters, we find a larger total magnetic moment(µJ ≈ 1.5 µB) than Stern-Gerlach results.

12

In addition to total magnetic moments, our XMCD re-sults also yield an orbital magnetic moment that is typ-ically ≤ 25 % of the spin magnetic moment for clustersin the size range considered here. This implies that anupper limit for the total magnetic moment per atom canbe estimated as the full atomic spin magnetic momentplus the reduced orbital contribution. Therefore, totalmagnetic moments > 5.0 µB per atom that have beenreported for small iron clusters167, and > 3.75 µB peratom that have been reported for cobalt clusters45 are indisagreement with our study and seem very unlikely.Consistent with our results and with the observed ab-sence of spatial alignment of the clusters in the ion trap,superparamagnetic behavior15 of 3d transition metalclusters is also found in Stern-Gerlach experiments onneutral 3d transition metal cluster beams14,46,169,172–178.

VI. CONCLUSION

In conclusion, spin and orbital magnetic moments ofiron, cobalt and nickel clusters were determined and dis-cussed in the size regime of n = 10 − 15 atoms percluster. In this size range Fe+13 has a significantly re-duced spin magnetic moment that cannot be observedfor Co+13 and Ni+13, as can be rationalized from the bulkphase diagrams. In other aspects iron, cobalt and nickelclusters behave quite similar: Except for Fe+13, all theseclusters are strong ferromagnets with completely filledmajority states and 1 µB spin magnetic moment per un-occupied 3d state. They are characterized by low mag-netic anisotropy energies of EMAE ≪ 65 µeV per atomfor iron and EMAE ≪ 52 µeV for cobalt and nickel clus-

ters, which leads to superparamagnetic behavior. Theaverage number of unoccupied 3d states in these clus-ters is nearly constant for a given element and is close tothe respective bulk value, even though the distributionof 3d holes over minority and majority states is differentfrom the bulk. The orbital magnetic moments of iron,cobalt, and nickel clusters are quenched to ≤ 25 % whilethe spin magnetic moments remain at 60 to 90 % of theatomic values. In comparison with existing DFT studies,we note a discrepancy between measured and calculatedmagnetic moments for iron, cobalt, and nickel clusters.The spin magnetic moments that are calculated withinDFT seem to be underestimated. In this context, theexperimental data of this work can act as a benchmarkfor theoretical studies on small and medium sized transi-tion metal clusters. Element specific XMCD of gas phaseions is a versatile technique that is not limited to puretransition metal clusters but will also allow the study ofalloys, compounds, oxides, or complexes.

ACKNOWLEDGMENTS

Beam time for this project was provided at BESSYII beamlines UE52-SGM and UE52-PGM, operated byHelmholtz-Zentrum Berlin. We thank T. Blume, S.Krause, P. Hoffmann, and E. Suljoti for technical assis-tance with the apparatus and during beam times. Thiswork was supported by the Special Cluster ResearchProject of Genesis Research Institute, Inc. and was par-tially funded by the German Ministry for Education andResearch (BMBF) under grant No. BMBF-05K13VF2.BvI acknowledges travel support by HZB.

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