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the great efforts devoted to the synthesis and investigation of nanosize magnetic molecules are motivated both by interests in fundamental physics and by the envisaged technological applications .
for instance , some of these systems have shown phenomena such as quantum tunneling of magnetization between quasi - degenerate levels , slow relaxation at low @xmath2 , and revealed to be promising for high density information storage and quantum computing @xcite .
the magnetic core of molecular magnets is constituted by transition metal ions sorrounded by an organic shell which prevents intramolecular magnetic interactions . as a result
, the microscopic properties of these nanoscale clusters can be investigated by means of bulk samples . among these systems , there are homonuclear antiferromagnetic ( af ) ring - shaped molecules formed by @xmath3 transition metal ions in an almost planar ring . in particular , in even membered rings
the dominant af exchange interactions lead to a singlet @xmath4 ground state and the energy spectrum is characterized by rotational bands , with the lowest - lying levels approximately following the so - called land s rule @xcite . in this paper
we study the magnetization and the phonon - induced relaxation in the heterometallic ring cr@xmath0ni .
this compound derives from the even membered af ring cr@xmath5 and thus provides a opportunity of a deeper insight in the role of topology in the static and dynamical quantum properties of magnetic wheels @xcite .
cr@xmath0ni compound is obtained by the chemical substitution of a cr@xmath6 ion with a ni@xmath7 ion in the structure of the cr@xmath5 ring .
this leads to a new molecular system formed by an odd number of unpaired electrons with dominant af nearest neighbour interactions as inferred by susceptibility measurements @xcite .
the resulting @xmath8 ground state has been shown to be suitable to encode a qubit @xcite .
the magnetic molecule has been theoretically analyzed within a spin hamiltonian approach , with the hamiltonian given by : @xmath9}\nonumber\\ + \sum_{i > j}\textbf{s}(i)\cdot\textbf{d}_{ij}\cdot\textbf{s}(j)-\mu_b\sum_i{g_{i}\textbf{h}\cdot\textbf{s}(i)},\end{aligned}\ ] ] where @xmath10 is the spin operator of the the @xmath11th ion in the molecule ( @xmath12=3/2 for cr@xmath6 ions , and @xmath12=1 for the ni@xmath7 ion ) .
the first term of the above equation is the dominant nearest neighbour heisenberg exchange interaction .
the second and the third terms describe the uniaxial local crystal fields and anisotropic intracluster spin - spin interactions respectively ( with the @xmath13 axis assumed perpendicular to the plane of the ring ) . the last term represents the zeeman coupling with an external field @xmath14 .
the parameters of the above hamiltonian were determined by inelastic neutron scattering ( ins ) experiments @xcite . in order to corroborate the microscopic description of the cr@xmath0ni from ins data ,
a detailed study of high field magnetization is very powerful .
in fact , with high pulsed fields up to almost 60 t , spin multiplets not accessible to a standard ins experiment can be explored . in fig.[fig : cr7nim - article]a the magnetization curve as a function of the magnetic field @xmath14 is reported . a clear staircase structure with plateaus at @xmath15 , @xmath16 and odd multiples of @xmath17 reflects the change in the ground state due to the external field at the level anticrossing fields @xmath18 .
an hysteresis of the measured magnetization curves has been observed .
the effect arises from the non - equilibrium condition due to the high pulsed magnetic field with a few millisecond duration @xcite and has been discussed in terms of phonon bottle - neck effects and magnetic foehn effects @xcite .
there is a very good agreement between the measured and calculated magnetization curves .
this is clearly visible in fig.[fig : cr7nim - article]b where the positions of the main peaks of the calculated and measured @xmath19 matches correctly .
the smaller peaks in the experimental @xmath19 are due to level anticrossings between excited energy levels .
the effects are caused by the non - equilibrium exeperimental conditions and are not included in equilibrium calculations reported in fig.[fig : cr7nim - article ] @xcite .
these results confirm that the microscopic picture derived from ins experiments @xcite perfectly holds even at very high applied magnetic fields .
ni ( top ) and derivative @xmath19 ( bottom ) at t=1.3k .
red ( dark gray ) and black lines represent the down and up experimental magnetic field processes respectively .
the dashed blue lines represent the theoretical calculation with the following parameters : @xmath20=16.9k , @xmath21=19.6k , @xmath22=-0.3k , @xmath23=-4k , @xmath24=1.98 , @xmath25=2.2.,scaledwidth=45.0% ]
a major obstacle to the proposed technological applications of magnetic molecules is constituted by phonon - induced relaxation .
in fact , molecular observables , e.g. the magnetization , are deeply affected by the interaction of the spins with other degrees of freedom such as phonons @xcite . here
we investigate the molecular spin - spin correlations through an approach based on a density matrix theory @xcite .
the irreversible evolution of the density matrix @xmath26 can be determined through the secular approximation and focusing on time scales detectable by low - frequency techiques such as nmr . within this theoretical framework , a general expression for the quasi - elastic part of the fourier transform of cross correlation functions
is given by @xcite : @xmath27 where @xmath28 is the dimension of the hilbert spin space of the molecule , @xmath29 is the equilibrium population of the @xmath30th level and @xmath31 , @xmath32 being the @xmath30th eigenstate of the spin hamiltonian , while @xmath33 is the so called rate - matrix .
the @xmath34 element @xmath35 of @xmath33 represents the probability per unit time of a transition between eigenstates @xmath32 and @xmath36 induced by the interaction of the spins with phonons . by assuming that spin - bath interaction takes place mostly through modulation of local crystal fields
, the rate matrix can be calculated on the basis of the eigenstates of molecular spin hamiltonian by first - order perturbation theory . with the choice of a spherical magnetoelastic ( me ) coupling @xcite the transition rates @xmath35 are given by : @xmath37 with @xmath38 , @xmath39 the gap between the eigenstates @xmath32 and @xmath36 of the molecule .
in the last equation @xmath40 are quadrupolar operators @xcite .
finally , @xmath41 represents the spin - phonon coupling strength , which can be determined by comparing the theoretical results with experimental data .
in fact , the nuclear spin - lattice relaxation rate @xmath1 probes the fluctuations of molecular observables , thus giving information on the relaxation dynamics @xcite . exploiting the moriya formula @xcite
, the proton nmr @xmath1 can be evaluated in absolute units using as inputs the positions of the cr and ni ions and of the hydrogens of the molecule : @xmath42 where the @xmath43 are the fourier transforms of the cross correlation functions from eq .
( [ eq : s ] ) calculated at the larmor angular frequency @xmath44 , while the @xmath45 are geometric coefficients of the hyperfine dipolar interaction between magnetic ions and protons probed by nmr .     for different values of the applied field along z ( parallel to the ring axis).,scaledwidth=45.0% ]    the occurence of a peak in the proton nmr @xmath1 has been clearly explained in homonuclear ring - shaped molecules with small anisotropy such as cr@xmath5 @xcite .
in fact , in this case @xmath46 , where @xmath47 is the fourier transform of the autocorrelation function of @xmath48 @xcite : @xmath49 $ ] .
this equation shows that the spectrum of fluctuations of @xmath48 is given by a sum of @xmath28 lorentzians , each with characteristic frequency @xmath50 , given by the eigenvalues of @xmath51 . for a wide range of @xmath14 and @xmath2 in these systems
only a single relaxation frequency @xmath52 significantly contribute to @xmath53 . as a result ,
if the dominant frequency @xmath52 intersects the larmor angular frequency , i.e. when @xmath54 , at the temperature @xmath55 the proton nmr @xmath1 shows a sharp peak @xcite .
being an heterometallic ring , this explanation does not hold for cr@xmath0ni and eq.([eq : t1 ] ) has to be used .
nevertheless , our calculations show that a peak in the reduced @xmath56 occurs in agreement with experimental data ( see fig.[fig : cr7ni - t1-hz ] ) . by fitting the observed peak position
we have obtained @xmath57thz@xmath58 .
a magnetization study of the heteronuclear antiferromagnetic ring - shaped nanomagnet cr@xmath0ni has been performed . a clear step - wise increase of magnetization with increasing field is observed .
the very good agreement of high field magnetization measurements up to almost 60 t with calculation shows the spin hamiltonian approach to be suitable even at very high fields .
the relaxation dynamics of the compound has been investigated by the proton nuclear - spin relaxation rate @xmath1 .
our calculations are in very good quantitative agreement with experimental data .
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we study the magnetization and the spin dynamics of the cr@xmath0ni ring - shaped magnetic cluster 
 . measurements of the magnetization at high pulsed fields and low temperature are compared to calculations and show that the spin hamiltonian approach provides a good description of cr@xmath0ni magnetic molecule . 
 in addition , the phonon - induced relaxation dynamics of molecular observables has been investigated . by assuming the spin - phonon coupling to take place through the modulation of the local crystal fields , it is possible to evaluate the decay of fluctuations of two generic molecular observables . 
 the nuclear spin - lattice relaxation rate @xmath1 directly probes such fluctuations , and allows to determine the magnetoelastic coupling strength .