Physica B 481 (2016) 224–231

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Nuclear spin-lattice relaxation at field-induced level crossingsin a Cr8F8 pivalate single crystal

Shoji YamamotoDepartment of Physics, Hokkaido University, Sapporo 060-0810, Japan

a r t i c l e i n f o

Article history:Received 4 September 2015Accepted 13 November 2015Available online 17 November 2015

PACS:75.50.Xx75.40.Mg75.75.-c76.60.-k

Keywords:Molecular ring antiferromagnetField-induced level crossingNuclear spin-lattice relaxationRecursion methodLanczos algorithm

x.doi.org/10.1016/j.physb.2015.11.01526/& 2015 Elsevier B.V. All rights reserved.

a b s t r a c t

We construct a microscopic theory for the proton spin-lattice relaxation-rate T1/ 1 measurements aroundfield-induced level crossings in a single crystal of the trivalent chromium ion wheel complex[Cr8F8(OOCtBu)16] at sufficiently low temperatures [E. Micotti et al., Phys. Rev. B 72 (2005) 020405(R)].Exactly diagonalizing a well-equipped spin Hamiltonian for the individual clusters and giving furtherconsideration to their possible interactions, we reveal the mechanism of T1/ 1 being single-peaked nor-mally at the first level crossing but double-peaked intriguingly around the second level crossing. Wewipe out the doubt about poor crystallization and find out a solution—intramolecular alternatingDzyaloshinsky–Moriya interaction combined with intermolecular coupling of antiferromagnetic char-acter, each of which is so weak as several tens of mK in magnitude.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

Observations of resonant magnetization tunneling in giant-spinclusters such as [Mn12O12(OOCMe)16(H2O)4] (Mn12 in short) [1,2] and[Fe8O2(OH)12(tacn)6]Br8 (tacn¼1,4,7-triazacyclononane¼C6H12(NH)3;Fe8 in short) [3,4] sparked broad and keen interest in molecular na-nomagnets. Much effort is still devoted to designing new varieties andexploring further applications in the context of nanomagnetism.Among numerous molecules of varied geometry is a class of ring-shaped clusters comprising an even number transition metal complexions, whose ground states are nonmagnetic unless magnetized ex-ternally, unlike those of Mn12 and Fe8, due to the dominant anti-ferromagnetic character of the nearest-neighbor interaction. While theepoch-making experiments [1–4] on ferrimagnetic clusters demon-strated quantum tunneling of the direction of the total spin [5–8], theantiferromagnetic ring clusters embody quantum fluctuation of themagnitude of the total spin [9–12]. Many of them have low-lying spinstates so closely spaced as to cross each other in a strong but attainablemagnetic field. It is a central issue for both experimentalists andtheoreticians to demonstrate whether, how, and why the crossingstates mix. An increasing field stabilizes higher-lying excited states oneby one into the subsequent ground state, where the crossing stateshave different parity. Neither local crystal field, whether axial orrhombic, nor external magnetic field, in any direction, causes repulsionbetween them [13–15]. It is challenging to probe into relevant mixingterms experimentally and to prove them theoretically [11,16]. Field-

induced level crossings in the antiferromagnetic ring clusters are thusinteresting and they can be monitored by various experimental tools.Here we focus on nuclear magnetic resonance (NMR), through whichwe can gain a microscopic insight into both quantum and thermal spindynamics [17–19].

Trivalent iron ion rings were of early interest for NMRinvestigations. Julien et al. [20] made a pioneering attempt toprobe the decanuclear ferric wheel [Fe10(OMe)20(mca)10] (mcaH¼monochloroacetic acid¼ClCH2COOH; Fe10 in short) with the protonspin-lattice relaxation rate T1/ 1, looking for the field-induced levelcrossings. As might be expected of a centrosymmetric structure, theFe10 ring exhibits cross relaxation, here in the sense of the twolowest-lying molecular energy levels coming so near as to matchwith the Larmor frequency, where T1/ 1 is strikingly enhancedwithout thermal assistance. However, available information on thequantum resonance was limited in the powder sample.

Then, in an attempt to analyze eigenstates mixing quantita-tively, single-crystalline trivalent iron ion rings were investigated.Proton T1/ 1 measurement [21] on the hexanuclear ferric wheel[LiFe6(OMe)12(dbm)6] ·BPh 54 CH2Cl2 (dbmH¼dibenzoylmethane¼1,3-diphenyl-1,3-propanedione¼CH2(OCPh)2; Li:Fe6 in short) withvarying field up to 15 T yields a bell-shaped broad maximum andits width, as large as 1.3 T, remains unchanged with decreasingtemperature. What can broaden the molecular levels so as to sa-tisfy the resonant energy condition over such a wide field interval?A trivial broadening due to the distribution of crossing fields is not

Fig. 1. Top (a) and side (b) views of a single molecular unit of [Cr8F8(piv)16].

S. Yamamoto / Physica B 481 (2016) 224–231 225

the case with the single-crystalline sample. Spin-phonon inter-action should be sensitive to temperature. Intermolecularinteraction, if any, could contribute far less than a tesla. We are ledto think of strong level repulsion of intrinsic origin. Heat-capacitymeasurements [21] of this sample indeed suggest substantialenergy gaps, 0.86 K and 2.36 K, at the first and second levelcrossings, respectively. Magnetic-torque measurements [11]of a sodium analog, [NaFe6(OMe)12(pmdbm)6]ClO4 [pmdbmH¼4,4′-(MeO)2dbmH¼1,3-di(4-methoxyphenyl)-1,3-propanedione¼CH2(OCC6H4OMe)2; Na:Fe6 in short] also reveal anticrossing gapsof good magnitude between the two lowest-lying states.

Proton NMR study [22] on the octanuclear ferric wheel[CsFe8(tea)8]Cl (teaH3¼triethanolamine¼N(C2H4OH)3; Cs:Fe8 inshort) disclosed still more striking level anticrossings. The gap atthe first level crossing, 2 K, amounts to one third of the zero-fieldsinglet-triplet gap, which should be compared with 0.86 K at thefirst level crossing to the singlet-triplet gap 15.7 K in Li:Fe6 (Ref.[21]). Such intense repulsion between magnetic levels stimulatedtheoreticians to lively discussions about the driving force. There isa possibility [23] of an intramolecular magnetoelastic instabilityinducing staggered Dzyaloshinsky–Moriya (DM) vectors [24],whereas another scenario [25] consists of intramolecular vibroniceffects on the local crystal fields in combination with an inter-molecular elastic interaction. In any case the level anticrossingdemands some structural perturbation to the regular polygon.

In such circumstances, we take further interest in obser-ving nearly pure level crossings in a high-quality sample. Theoctanuclear chromic wheel [Cr8F8(piv)16] (pivH¼pivalic acid-¼trimethylacetic acid¼tBuCOOH; Cr8 in short) was found to meetthe demand. Heat-capacity measurement with increasing mag-netic field [26] on the sample in a tetragonal crystalline form [27]evidences very little repulsion between the two lowest-lyingstates. The first two crossings monitored by 1H NMR indeed ex-hibit narrower maxima of T1/ 1 than those in Li:Fe6 [28]. Observa-tions in more detail [29] look still more intriguing: T1/ 1 is single-peaked at the first level crossing but double-peaked around thesecond one. It is quite rare to find the proton spin-lattice relaxa-tion of molecular ring antiferromagnets being accelerated in sucha complicated manner at level crossings. The proton T1/ 1 ob-servations [20] of Fe10 are obtainable up to 15 T, that is, beyond thethird crossing field, but each structure, if any, could not be resolvedwithin the use of powder samples. Although the proton T1/ 1

measurements [21] of Li:Fe6 single crystals range over the samefield, they only cover the first level crossing due to stronger in-tramolecular exchange coupling. The detailed proton NMR study[22] on Cs:Fe8 single crystals are also restricted to the first levelcrossing. We are thus eager to solve this unusual structure of T1/ 1

in a Cr8 single crystal.

2. Modeling and numerical method

The cyclic cluster Cr8 in issue [27] consists of trivalent chromiumions of spin 3

2forming an almost planar octagon, as is depicted in

Fig. 1. We analyze it through a well-equipped Hamiltonian,

= + + + + ( )– ; 1ex ligand Zeeman dipolee e

DM

∑= ·( )=

+s sJ ,2ai

i i iex1

8

1

⎡⎣ ⎤⎦∑ ∑= ( ) + ( ) − ( )( )= =

d s e s s ,2bi

i ic

ii i

aib

ligand1

82

1

82 2

∑μ= − ·( )=

H sg ,2ci

iZeeman B1

8

⎡⎣⎢⎢

⎤⎦⎥⎥∑=

·−

( · )( · )

( )–

≠

s s s r s rl Ir r2

3,

2di j

i j

ij

i ij j ij

ijdipolee e

3 8

3 5

∑= ·( × )( )=

+D s s .2ei

i i iDM1

8

1

where the Cr3þ spins are numbered in a cyclic manner ( =+s si i8 ).

ex is the intramolecular isotropic Heisenberg exchange interactionof antiferromagnetic character ( >J 0i ). ligand describes the localcrystal fields of the axial and rhombic types, referring to the crys-tallographic axes as ( )a b c, , , the last of which corresponds to themolecular symmetry axis. Cr8 is not an ideal ring. Its point sym-metry is D4d at the highest [30,31] and with decreasing tempera-ture, the second-order in-plane anisotropy possibly comes into ef-fect [32] at the cost of the uniaxial four-fold rotation [33,34]. A

S. Yamamoto / Physica B 481 (2016) 224–231226

deformed ring may have bond-dependent exchange and/or site-dependent crystal-field parameters. We have indeed tested varia-tions of Ji, di, and ei to see if they would affect T1/ 1 especially nearlevel crossings, but they do not have any significant effect. Neitherthe magnetic-susceptibility [27] nor the inelastic-neutron-scatter-ing spectrum [32] is better reproduced in consideration of alter-nating Heisenberg exchange and/or crystal-field parameters. Thus,discarding the bond and site indices of these parameters,we adopt the estimates =J k/ 17 KB , = −d k/ 0.34 KB , and

= −e k/ 0.046 KB . Zeeman represents the Zeeman interaction with auniform external magnetic field whose direction defines thequantization axis z, where we assume the isotropic g factor to be 2.

dipole is the intramolecular dipolar interaction with μ= ( )I g l/B2 3

and =→

r ijij , where the average nearest neighbor distance

≡ ∑ = +l r /8i i i18

, 1 measures 3.388 Å (Ref. [27]). DM brings neighbor-ing spins into antisymmetric interaction [35,36] compatibly withthe fourfold symmetry for cyclic permutations. Effects of such anantisymmetric term were discussed in detail in an attempt to in-terpret the substantial energy gaps at level crossings in the hexa-nuclear [11,21,37,38] and octanuclear [22–24] ferric wheels. Whenwe decompose the DM vectors Di into their net andalternating components as ¯ ≡ ( + + )− +D D D D2 /4i i i i1 1 and

≡ ( − − )͠ − +D D D D2 /4i i i i1 1 , the former contribute nothing to a levelanticrossing, while the latter are able to mix the lowest-lying statesinvolved in the level crossing and to mix states of different parity inmore general, subject to deformation of the ideal ring [11,24,38].Hence we consider alternating DM vectors parallel to the c axis,

≡ ( ) = ( − ) ( )D D D D D, , 1 0, 0,i ia

ib

ic i . Chromic wheels have hardly

been viewed in this light so far in the belief that their lowest-lyingmagnetic states would mix little in the ground-state level crossings.Considering that the cyclic permutation symmetry of Cr8 is 4 or lessat low temperatures, possible DM vectors may alternate along thering and they, however small in magnitude, can have an effect of nolittle significance on the NMR observations in cooperation withsome other interactions.

A proton in the pivalate ligand is coupled with the Cr3þ ions ofthe ring through the dipolar interaction

∑= ℏ ( + + )( )

–

=

+ − − +F I F I F I ;3i

i i iz z

dipolee n

1

8

= − + + ( )± ( ) ± (∓ ) (∓ ) ∓F D s D s D s , 4ai i i i iz

i i16

0 1 2

= + + ( )( ) (+ ) + (− ) −F D s D s D s , 4bi

zi i

zi i i i

23

0 1 1

where the geometric factors are given by

γ γ Θ= ( ℏ )( − ) ( )( )D R3 /2 3 cos 1 , 5ai i i0

e p3 2

γ γ Θ Θ= ( ℏ ) ( )Φ(± ) ∓D R e3 /2 sin cos , 5bi ii

i i1

e p3 i

γ γ Θ= ( ℏ ) ( )Φ(± ) ∓D R e3 /4 sin , 5ci ii

i2

e p3 2 2i

with the spherical coordinates of site i to the probe proton,Θ Φ( )R , ,i i i , and the electron and proton gyromagnetic ratios,

γ μ≡ ℏ = ×g / 1.758 10 rad Hz/Te B11 and γ ω≡ = ×H/ 2.675p p

10 rad Hz/T8 , respectively. The longitudinal relaxation of a protonnuclear spin reads as

∫ ∫δ δ δ δ= ⟨ ( ) ( )⟩ + ⟨ ( ) ( )⟩( )

ω ω−∞

∞− + −

−∞

∞+ −

TF t F e t F t F e t

10 d 0 d ,

6i t i t

11 1 1 1

p p

considering that the spin- si contribution is proportional to −Ri6,

where ⟨ ⟩ means taking the canonical ensemble average andδ ( ) = − ⟨ ⟩± ℏ ± − ℏ ±F t e F e Fi t i t

1/

1/

1 (Ref. [39]). Under such conditions thatneither the square nor the z component of the total spin∑ ≡= s Si i1

8 commutes with the full Hamiltonian , Cr8 is too hardto fully diagonalize [40–42]. At moderately low temperatures,

ωℏ ⪡ ⪡k T Jp B , the thermal average (6) can be reduced to

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

∑π δ δ ω

ω

= ⟨ | | ⟩ − −ℏ

= − +ℏ

+ ϵ( )δ

−

ϵ→+−

Tn F

E E

GE

i

14 0

4 lim Im ,7

n

n

F

11

2p

0

0p

01

with | ⟩n being the eigenvector of energy En ( ≤ ≤ ≤ …E E E0 1 2 )for the Hamiltonian (1). The Green function ( )≡δ

+−G zF1

δ δ⟨ | ( − ) | ⟩+ − −F z F0 011

1 can be evaluated as a continued fraction,

δ δ

αβ

αβα

( ) = ⟨ | | ⟩

− −− −

− ⋯ ( )

δ+−

+ −G z

F F

z

zz

0 0,

8

F1 1

112

222

3

1

where αn and βn are obtained by a recursion method based on theLanczos algorithm:

( )α β β| ⟩ = | ⟩ − | ⟩ − | ⟩ ( )+ − −f f f f / ; 9an n n n n n n1 1 1

α = ⟨ | | ⟩ ( )f f , 9bn n n

β α β= | ⟩ − | ⟩ − | ⟩ ( )− −f f f , 9cn n n n n n1 1

with the initial vector | ⟩f1 and coefficient β0 being defined asδ δ| ⟩ || | ⟩||− −F F0 / 01 1 and 0, respectively.

3. Single-cluster calculation

The proton T1/ 1 of Cr8 was measured up to 15.6 T at 1 K and 0.3 K(Ref. [29]). Eq. (7) is valid enough for the 0.3 K observations and isLorentzian-broadened so as to fit inwith them. In accordance with theexperiment, we tilt the external magnetic field to the molecularsymmetry axis c at an angle of thirty degrees unless otherwise noted.In the tertiary butyl (tBu) group of each pivalate ligand lies the nineprotons. Half of the sixteen pivalate ligands bridge the Cr3þ ionsaround the outside of and almost in-plane with the wheel [Fig. 1(a)],while the rest of them do alternately above and below the plane atnearly right angles [Fig. 1(b)]. With an arbitrary field being applied,there are many inequivalent proton sites extrinsically as well ascrystallographically. To complicate matters still further, all the tBugroups are highly disordered and the pivalate ligands out of the wheelplane are disordered over two positions [32]. Consequently, the re-covery of the nuclear magnetization is far from single-exponential.Then it is an idea to regard the time for which the magnetizationrecovers to e1/ of the equilibrium value as a sort of average T1 over allthe inequivalent protons [29]. In order to calculate the geometricfactors accordingly, we consider “pruning” the ligand molecule [30],that is, replacing every disordered methyl molecule CH3 with an iso-valent H atom. This is a procedure which they call “hydrogen satura-tion” and it alters neither the magnetic core nor the point symmetry ofthe present molecule. Combined torque and heat-capacity measure-ments [26] of a [Cr8F8(piv)16] single crystal are in good agreement withab initio calculations [30,43] of a fully pruned Cr8 ring. Fig. 2 demon-strates how the pivalate ligand is pruned. In the fully pruned moleculeCr8F8(OOCH)16, we have a single H atom instead of each tBu group and

Fig. 2. Pruning procedures for the pivalate ligand: (a) turns into (b) and ends upwith (c).

Fig. 3. T1/ 1 versus H for Cr8. Experiment for a single crystal at 0.3 K (Ref. [29]) andtheoretical findings with no DM interaction. (a) Calculations for a single moleculewith its symmetry axis c crossing the field at various angles θ. (b) Calculations for atwin sample with θ being half-and-half. (c) Calculations for a polycrystallinesample with a random distribution of θ.

S. Yamamoto / Physica B 481 (2016) 224–231 227

all the protons, whether in or out of the wheel plane, are equallydistant from the nearest Cr3þ ion. When we set the dipolar couplingconstant γ γℏ R/e p 1

3 equal to ×3.9 10 rad Hz6 , which means that

≃ ˚R 5 A1 and agrees well with experimental findings [27,29], the thus-calculated T1/ 1 successfully reproduces the observations.

First of all we inquire into a possibility of the sample beinggeminate or mosaic. Since the experimental investigations [28,29]were self-contained in such a way that a close X-ray-diffractionanalysis of the crystal gave no indication of twinning, we are neverafraid of such a trivial scenario a different crossing field for eachsingle crystal could cause some impure maximum in T1/ 1 versus H.We wipe away any such concern from the theoretical point ofview. Tilting the field little by little, we calculate T1/ 1 as a functionof the field strength for the Cr8 regular octagon. The calculationsare compared with the observations [29] in Fig. 3(a). In the NMRexperiment the authors infer from the comparison of their find-ings with separate torque measurements [27] that the angleθ = °29.5 . The theoretical findings for the first level crossing in-deed fit the observations well when we set θ equal to 30°. Thecrossing field reaches a minimum value at every level crossingwhen the magnetic field is perpendicular to the crystal field axis.In an attempt to reproduce the double-peaked enhancement at thesecond level crossing, we superpose, for instance, the calculationsat θ = °0 upon those at θ = °30 in Fig. 3(b). It is not the secondmaximum but the first one that splits into two peaks. The crossingfield is much less sensitive to θ for higher-lying multiplets. Fig. 3(c) shows that even powders with randomly oriented crystal fieldaxes could not exhibit such a wide-split maximum at the secondlevel crossing. Hereafter, we fix θ at 30°, unless otherwise noted,and search for an intrinsic mechanism to yield such intriguingobservations of T1/ 1.

Now we consider introducing a DM term into the Hamiltonian.Fig. 4 shows that every T1/ 1 maximum turns into twin peaks withstaggered DM vectors growing. While the split may be hard todetect at the first—singlet–triplet—level crossing, it should surelybe recognizable at those between higher-lying levels—triplet–quintuplet, quintuplet–septuplet, and so on. It is also convincingthat the higher-field level crossing, the lower maximum. Some offerric wheels are well known for their level anticrossings and theyare describable with substantial DM interaction. The amplitude ofthe alternating DM vectors, D, reads a few hundred mK for Na:Fe6(Ref. [11]) and Cs:Fe8 (Ref. [24]) or more for Li:Fe6 (Ref. [11]). Thepresent estimate of D k/ B is no more than 0.068 K. No wonder sucha weak interaction was unnoticed in inelastic-neutron-scatteringmeasurements at 2.5 K [32].

4. Single-crystal calculation

We have obtained a double-peaked structure of T1/ 1 around thesecond crossing field, but it leaves much room for improvement.With increasing DM interaction, each maximum splits in two, buttheir center keeps still. When we define a critical field relevant tothe crossing between multiplets of spin S and +S 1, ( + )H S S, 1c , as[ ] == ( + )S Hd /d 0z

H H S S2 2

, 1c , the twin peaks sit across Hc from eachother (cf. Fig. 7). Even if we further turn D up, the twin peaksshould come apart more and more leaving Hc untouched. Theproduct would be nothing more than an ill-located mountain with

H ( T )614186

T 1−1

(sec−1

)

100

10−2

= 2 10−3

I' / J = 0

D / JD / J = D / J =

experiment

J' / J = 0

4 10−3

6 10−3

T 1−1

(sec−1

)

100

10−2

0201

D / J = 4 10−3

I' / J = 0J' / J = 0

experiment

H ( T )Fig. 4. T1/ 1 versus H for Cr8. Experiment for a single crystal at 0.3 K (Ref. [29])compared with calculations for a single molecule under varying DM interaction(a) and one of the above calculations presented in a wider range of the field (b).

H ( T )614186

0201H ( T )

T 1−1

(sec−1

)

100

10−2

D / J = 4 10−3

I' / J = 5 10−5

J' / J = 3 10−3

T 1−1

(sec−1

)

100

10−2

J' / J = 0J' / J = J' / J =

D / J = I' / J =

4 10−3

5 10−5

2 10−3

4 10−3

experiment

experiment

Fig. 5. T1/ 1 versus H for Cr8. Experiment for a single crystal at 0.3 K (Ref. [29])compared with calculations for a single crystal with tetragonal packing undervarying intermolecular antiferromagnetic interaction (a) and the best solutionpresented in a wider range of the field (b).

S. Yamamoto / Physica B 481 (2016) 224–231228

a wide crater on the top. The discrepancy looks more significantfor higher-lying magnetic levels. Each of the twin peaks arisesfrom a different magnetization background. The magnetizationincreases by one every time we stride across Hc. There must besomething to push up Hc in proportion to the ground-statemagnetization.

Thus and thus, we finally incorporate an intermolecular cou-pling effect into our theory. Here we assume hydrogen bonding aswell as dipolar coupling to bring clusters into interaction. Themeasured Cr8F8 pivalates assemble into the tetragonal space groupP4212 [27], with a¼b¼20.09 Å and c¼16.80 Å at room tempera-ture, where the molecules are aligned in a parallel fashion andeach unit cell contains four halves of them. The tetragonal packingis tighter than a monoclinic crystalline form of an early sample[44]. The ring–ring dipolar energy is about half mK, which is twoorders of magnitude smaller than D and thus of little significancein modifying the present level scheme, while there is a demon-stration in a similar system [45,46] of hydrogen bonds mediatingappreciable intermolecular antiferromagnetic superexchangecoupling of the order of μ10 eV . Considering that half of the pi-valate ligands lie around the outside of the chromic wheel and therest of them rise from the wheel plane, we are led to expect asuperexchange pathway through intermolecular hydrogen bondsto stabilize the crystal packing [47].

Within the first-order perturbational treatment both interac-tions can be described in terms of the cluster total spins

≡ ∑ =S si i18 . Labeling the molecule under our microscopic ob-

servation, which is exactly diagonalized, and its surroundings,which work as if they are a mean field, as n and m, respectively,we express the intermolecular interactions as

′ = ′ + ′ ( ); 10ex dipole

⎪ ⎪

⎪ ⎪⎛⎝⎜

⎞⎠⎟

⎧⎨⎩

⎫⎬⎭

∑

( )μ

′ = | + | ′¯ ·

| − |− [ ¯ ·( − )][ ·( − )]

| − |

≡ − ′ · 11a

a b S Sm n

S m n S m nm n

H S

I

g

23

,

m

m n m n

n

dipole

3

3 5

B dipole

∑ μ′ = ′ ¯ · ≡ − ′ ·( )

S S H SJ g ,11bm

nm m n nex B ex

where the mean-field variables ¯ ≡ ⟨ ′| | ′⟩S S0 0m m are self-consistentlydetermined, so as to give a convergent ground state, | ′⟩0 , in the“effective field” + ′ + ′H H Hex dipole. The dipolar coupling constant ′Ireads as μ( ) ( )g a2 /B

2 3 for the tetragonal lattice, whereas the an-tiferromagnetic superexchange coupling constants ′ mJn are setfinite, referred to as ′J , only for the nearest neighbors m. ′J is takenmoderately within the order of few μeV.

Fig. 5 shows that the intermolecular interactions significantlyimprove the calculations. Since the molecular field ′ + ′H Hex dipole

reduces the external field H , every maximum of T1/ 1 shifts right.Since the molecular field ′ + ′H Hex dipole grows with increasingground-state magnetization, a higher-field maximum moves more.The calculations remain unchanged until the ground state ismagnetized. The first maximum, its left-hand (nonmagnetic) sidein particular, is thus little affected. The second one, arising from atriplet-to-quintuplet crossing, splits wider and moves rightward abit more. It is all the more noticeable with the third one, arisingfrom a quintuplet-to-septuplet crossing [cf. Figs. 4(b) and 5(b)].When we tilt the field more, we expect wider-split maxima of T1/ 1,as is demonstrated in Fig. 6(a), and such measurements couldensure the present scenario. The twin peaks are most impressivewhen the field is perpendicular to the molecular symmetry axis.Fig. 6(b) leads us to better understanding of these observations.Location of the critical field giving a maximum susceptibility is

T 1− 1

(sec−1

)

100

10−2

0201H (T )

θ = 300

θ = 600

θ = 900

D / J = 4 10−3

I' / J = 5 10−5

J' / J = 3 10−3

0.0

1.0

2.0

3.0

S

D / J = 4 10−3

I' / J = 5 10−5

J' / J = 3 10−3

θ = 300

θ = 600

θ = 900

Fig. 6. Calculations of T1/ 1 (a), together with Sz (b), versus H for a single crystal withtetragonal packing under varying direction of the field H .

E n /J

−21.0

H (T)20151050

0.0

S

D / ' / J

4 10−3

4 10−3 5 10−5 3 10−3

J'/ J0 0 0

0 0

J I

1.0

2.0

3.0

7 8−23.10

−23.18

E0

/J

15 2241 21

−24.80

−24.70

D / '/ J

4 10−3

4 10−3 5 10−5 3 10−3

J'/ J0 0 0

0 0

J I

H (T)

−22.0

−24.0

−25.0

100

10−3

10−2

101

T 1−1

(sec−1

)

H (T) H (T)

−23.65

−23.70

E 1/J

Fig. 7. Calculations of the low-lying energy levels En (a), the proton spin-latticerelaxation rate T1/ (b), and the ground-state magnetization Sz (c) as functions of H

S. Yamamoto / Physica B 481 (2016) 224–231 229

sensitive to the field direction for the first level crossing but ismuch less so for the following. In the case of both the levels in-volved being magnetic states, the field direction has a major effecton ranging of the resonance field allowing nearly cross relaxationbetween the electronic and nuclear reservoirs [20]. Magneticstates mix well over a wider field interval with increasing θ. Thelow-temperature torque-magnetometry measurements [26] in-deed reveal significant S mixing.

In our theory a wide-split maximum of T1/ 1 does not necessarilycome from a large anticrossing gap. Our theory neither conflictswith the heat-capacity measurements at 0.9 K (Ref. [26]) claiminga vanishingly small anticrossing gap and thus weak DM interactionin comparison with those in some ferric wheels [11,21,38,22–24],nor prevents the inelastic-neutron-scattering measurements at2 K (Ref. [32]) from indicating that intermolecular interactions arenegligible. The broad double-peaked maximum of T1/ 1 withoutthermal assistance at level crossings between magnetic states isthe fruit of cooperating intramolecular alternating DM interactionand intermolecular coupling of antiferromagnetic character, bothof which are no more than several tens of mK in magnitude.

1for a single Cr8 molecule without ( = ′ = ′ =D I J 0) and with ( ≠ ′ = ′ =D I J0, 0) al-ternating DM interaction and for a tetragonally packed Cr8 single crystal with in-tramolecular alternating DM interaction and intermolecular antiferromagnetic su-perexchange coupling ( ≠ ′ ≠ ′ ≠D I J0, 0, 0). Level crossings between the two low-est-lying states are enlarged.

5. Discussion and future aspect

Fig. 7 reviews the present scenario in terms of the energy levelscheme. The intramolecular dipolar interaction does not con-tribute in itself to the gap formation at each level crossing (see thecase of = ′ = ′ =D I J 0) unless it breaks the centrosymmetricstructure. The DM interaction makes the crossing levels repulsiveto each other keeping the critical field Hc unchanged (see the caseof ≠D 0 and ′ = ′ =I J 0). The thus-induced first [ Δ( )0, 1 ], second[Δ( )1, 2 ], and third [Δ( )2, 3 ] anticrossing gaps are 0.094 K, 0.20 K,and 0.30 K, respectively. Micotti et al. [29] measured the proton

T1/ 1 of single-crystalline Cr8F8 pivalates as a function of tempera-ture as well as magnetic field in an attempt to trace the antic-rossing gap. The relaxation is thermally activated even in closevicinity of the first critical field ( )H 0, 1c and the activation energyreads 0.1070.05 K, which is in excellent agreement with our es-timate of Δ( )0, 1 . This is, on the other hand, somewhat larger than

another experimental—magnetocaloric—estimate [26], but theauthors themselves [26,29] claim that the two sets of findingsshould be considered consistent in view of the large uncertainty inboth measurements. In any case the level anticrossing gaps in Cr8are about one order of magnitude smaller than those in Li:Fe6,Δ( ) =0, 1 0.86 K and Δ( ) =1, 2 2.36 K (Ref. [21]), and such smallgaps may be hard to evaluate in near-kelvin measurement [26,32].The intermolecular interactions hardly affect the anticrossing gapΔ but push up the critical field Hc. The change in Δ( + )S S, 1 is nomore than 1 mK even for the third crossing S¼2. The DM inter-action is decisive of a maximum extent of the S mixing, while theintermolecular superexchange interaction widens the S-mixed

S. Yamamoto / Physica B 481 (2016) 224–231230

field range. Since each eigenvalue moves upward in proportion toits magnetization Sz times the effective ground-state magnetiza-tion S̄ z, the resonance condition for the electronic and nuclearreservoirs is met over a wider field interval. The rightward shiftand increasing split of every T1/ 1 maximum with growing inter-molecular coupling is thus achieved.

In our theory the first maximum of T1/ 1 may have hyperfinestructure, subject to relevant intermolecular coupling, while the ex-perimental resolution is insufficient at 0.3 K. Considering that theanticrossing gap Δ( )0, 1 is no more than 0.1 K but still one order ofmagnitude larger than the nuclear Zeeman level splitting, the nuclearspin-lattice relaxation yet remains phonon-assisted in sub-kelvintemperatures. Indeed, at 1 K, the striking enhancement of T1/ 1 mod-erates and the twin peaks of the second maximum almost melt away[29]. If we discuss the first level crossing of Cr8 in more detail, we haveto go down to lower temperatures in a laboratory. Micotti et al. [29]have pointed out that the first level crossing involves a nonmagneticstate, whereas in the second level crossing both the states involved aremagnetic, and this can change the nuclear magnetic relaxation me-chanism, which hits the key ingredient of our theory. It is interesting[41] from this point of view to observe the first level crossing in theheterometallic analogs [{R2NH2}{Cr7MF8(piv)16}] (R¼alkyl chain¼CnH2nþ1; M¼Mn, Fe, Co, Ni; Cr7M in short) [48]. The substitution of adivalent transition metal +M2 for a Cr3þ ion induces an extra spin inthe otherwise fully compensated octanuclear cyclic molecule; thenboth the states involved in the first level crossing are magnetic andintermolecular interactions, if any, must come into effect. Sizable DMvectors originating in the two Cr-M bonds [49] can also contribute toany further observations of S mixing, though full calculation for theextended Hamiltonian (1)þ (12) without any cyclic permutationsymmetry is awfully time-exhausting.

We are in addition eager to observe the third level crossing, whichmay, however, be infeasible with proton nuclei in the present sample[Cr8F8(piv)16], where the relevant Hc lies beyond 20 T. It is interestingin this context to replace the bridging fluorides and/or pivalates byother molecules. The hydroxo analog [Cr8(OH)8(piv)16] (Ref. [50]) has aweaker exchange coupling, =J k/ 14.5 KB , where the third levelcrossing is attainable with a smaller field. Another hydroxo analogwith acetate bridges, [Cr8(OH)12(OOCMe)12] [51], looks still more at-tractive. Each of the two neighboring Cr3þ ions is bridged either bytwo hydroxo and one acetato ligands, with a Cr ··· Cr distance,

= ˚r 2.949 A12 , and an antiferromagnetic exchange coupling,=J k/ 13.7 K1 B , or by two acetato and one hydroxo ligands, with

= ˚r 3.383 A23 and =J k/ 8.9 K2 B , in an alternating fashion. The low-lying eigenvalues of the Hamiltonian (1) are insensitive to the alter-nation and the crossing fields are nearly equal to those for a regularoctagon with the mean coupling ( + )J J /21 2 . We may expect the thirdlevel crossing within 15 T. The alternating ligand molecules may causethe Cr8 ring to deviate further from a planar as well as regular octagon,leading to an enhancement of the DM interaction. In addition, thecomplex [Cr8(OH)12(OOCMe)12], together with 34 H2O, crystallizes in atetragonal lattice with an extensive network of hydrogen bonds. Fur-ther derivative with phenylacetate bridges, [Cr8(OH)8(OOCPh)16] (Ref.[52]), is also intriguing in this context. Aromatic molecules are welldelocalized and apt to contribute toward intermolecular hydrogenbonding [45,46]. We hope that a comparative study on these familycompounds will serve to verify the relevance of the intermolecularsuperexchange coupling as well as the intramolecular alternating DMinteraction to the split maxima of T1/ 1.

Acknowledgments

We thank Y. Furukawa, A. Lascialfari, and F. Borsa for theirencouraging us in our studies and enlightening us on their

experiments. We are further grateful to Y. Noriki for his help indrawing figures. This study was supported by the Ministry ofEducation, Culture, Sports, Science, and Technology of Japan.

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