Thermally activated and field-tuned tunneling in Mn 12 Ac studied by ac magnetic susceptibility

The magnetic ac susceptibility x of oriented Mn12Ac crystallites has been measured as a function of temperature, field, and frequency. The field has been applied at different values of the angle u with respect to the sample easy axis. For T55 K, the isothermal and adiabatic x limits have been determined as a function of field. Foru50° and intermediate frequencies, Lorentzian-shaped peaks have been observed at magnetic field valuesHn5nH1 with n50, 1, and 2 whereH154.1 kOe. Asu increases these maxima shift to higher fields, that satisfyHncosu5const, and decrease in amplitude. The relaxation time 1 follows Arrhenius’ law with respect to temperature and decreases sharply at H5Hn . The observed phenomenology unambiguously proves the existence of field-tuned tunneling between excited magnetic states which are thermally populated. At 5 K, the effective activation energy and the spin states involved in the tunneling process have been obtained. @S0163-1829 ~97!09317-X#


I. INTRODUCTION
Since the report on the Mn 12 Ac complex magnetic bistability and the possibility that this molecule provides a magnetic quantum tunneling 1 ͑MQT͒ model system, extensive work has been performed on it.Magnetic dc susceptibility experiments characterized Mn 12 Ac as a superparamagnet, with blocking temperature T B ϭ3 K.It was found that this compound has only one relaxation time , which depends exponentially on the temperature ϭ 0 exp(QV/k B T), with 0 ϭ2.1ϫ10Ϫ7 s and QV/k B ϭ61 K.The saturation of the relaxation time to a constant value below 2 K was interpreted as a new indication of the crossover to MQT below that temperature. 2he bistability property was based on the appearance of a broad magnetic hysteresis loop below T B . 1 Later, ''avalanche'' processes appeared in hysteresis loops measured below 2.5 K. 3 Recently [4][5][6] hysteresis loop measurements on a sample of oriented crystals showed the onset of steps at field intervals of ⌬Hϭ4.6 kOe, which were attributed to resonant tunneling of the magnetization between different excited quantum states.This process differs from MQT between the lowest-lying energy states for the two magnetization directions.However, both processes are related since the MQT is the mechanism which gives rise to magnetic relaxation of the molecule.
A well-established technique to study magnetic relaxation times is ac susceptibility.Indeed, the complex susceptibility was studied on this molecule as a function of frequency and temperature, confirming the values of QV/k B and 0 reported earlier. 7Of more interest is the field dependence of the relaxation time measured by means of ac susceptibility. 8In Ref. 8 it is described that when the magnetic field is applied on an oriented sample along the c axis, the relaxation time shows two dips at Hϭ0 and Hϭ3 kOe, respectively.The authors inferred the existence of tunneling between thermally populated up, ͉ϩm͘, and down, ͉Ϫm͘, magnetization states, which are nearly degenerate at zero field.When a field is applied, the increase in could be related to the detuning produced by the Zeeman splitting of the ͉Ϯm͘ doublet.A relaxation time minimum would appear when energy levels corresponding to states of opposite spin orientation cross and a new tuning condition is achieved.
Although the value H c ϭ3 kOe is different from ⌬H ϭ4.6 kOe we may conjecture that the step in magnetization and the dip in are simply related.To ascertain this, ac susceptibility measurements have been performed on the same sample for which the steps had been previously detected.The magnetic field, frequency, and temperature dependence of the complex susceptibility have been explored in the neighborhood of the tuning regions ͑Hϭ0 and 4.6 kOe͒, as a function of sample orientation.Briefly, the presence of susceptibility peaks at the tuning fields, with a Lorentzian dependence on the applied field, has been observed.From the temperature and field dependence of the relaxation time, the existence of tunneling between thermally populated excited levels at these fields has been unambiguously concluded.The relaxation time depends on the orientation of the easy axis of the sample with respect to the applied magnetic field.

II. SAMPLE PREPARATION AND EXPERIMENT
The compound Mn 12 Ac, brief for is an organometallic molecular crystal with 8 Mn III (S ϭ2) and 4 Mn IV (Sϭ3/2) ions, which form a magnetic cluster of effective spin Sϭ10 ͑Ref.9͒.The sample was synthesized as described in Ref. 10 and its quality was checked by x-ray diffraction.The obtained crystallites were about 10 m long and aspect ratio of about 10, the long axis being parallel to the c axis.The crystallites were embedded in Araldite epoxy and submitted to a 5.5 T field at room temperature.Since at this temperature the sample is superparamagnetic, an effective orientation of the crystallites with the c axis parallel to the field direction was achieved.The orientation was confirmed visually under a microscope at 1000ϫ magnification.We estimate that the crystallites' c axis lie within a cone of about Ϯ7°wide, around the field direction.
Cylindrical samples of this sample/epoxy solid were measured in a Quantum Design superconducting quantum interference device magnetometer with ac susceptibility option.The excitation field was 4 Oe for all experimental runs.The frequency was varied between 0.025 and 980 Hz and the temperature stability was maintained within 1% of the absolute temperature.Measurements were also performed with the orientation axis ͑taken as the z axis of the sample͒ forming an angle 0ϽϽ90°with the applied field.For this the sample holder was cut to form the desired angle and the sample glued on that surface.We estimate that the angle between the sample axis and the field direction was determined to 5°.

III. EXPERIMENTAL RESULTS
The ac susceptibility measured under a 5 Hz ac magnetic field applied along the z axis of the sample is shown in Fig. 1.As it has been recently reported by several authors 1,2,8 it follows the characteristic behavior of an ideal superparamagnetic system with a well defined relaxation time.Between 6 and 12 K the total susceptibility is real and follows the Curie-Weiss law ϭC/(TϪT C ) whereas it reaches a maximum at the blocking temperature T B ϭ5.5 K and decreases at lower temperatures.The imaginary part of the susceptibility Љ departs from zero near T B and shows a peak at T ϭ4.5 K.At this temperature, the relaxation time 1 of the Mn 12 Ac molecules equals the inverse of the angular frequency (ϭ2) of the ac magnetic field.On the other hand, the characteristic measuring time of dc magnetic experiments is about 100 s and, consequently, the dc blocking temperature T B ϭ3 K, obtained from the peak of the dc zero-field-cooled magnetization curve, 1 and the value reported here are very different.
It should be noted that two shoulders appear in Ј(T) and Љ(T) below 4 K.These two peaks are related to relaxation processes with relaxation times 2 and 3 , respectively, shorter than 1 .The existence of these processes has also been inferred from magnetic relaxation experiments performed below 3 K ͑Ref.2͒.Our Ј͑͒ measurements performed above 4 K reveal that these faster relaxing parts of the magnetization contribute to Ј with their equilibrium susceptibility for the whole measuring frequency and dc field range.Then, it follows that 1 ӷ 2 ӷ 3 and we focus our study on the slowest relaxation mechanism.
Our ac susceptibility measurements under a dc field have been performed at TϾ3 K. Furthermore, the magnetization reaches the thermal equilibrium value M eq ; consequently, magnetic relaxation effects are only due to the ac magnetic field.For practical purposes, the frequency-dependent susceptibility can be written as follows: 11 ac ͑ H,T,, where is the angle of the applied dc and ac magnetic fields with respect to the easy axis of the sample.Usually, the dynamic behavior of an uniaxial single domain magnetic particle is analyzed using Arrhenius' law for the relaxation time: 12,13 where the energy barrier U(H,) for thermal activation, in the case of uniaxial magnetic anisotropy, is given by 14,15 U͑H, ͒ϭQV͓1ϪH/H Q ͔͑͒ x͑͒ .͑3͒ Here Q is the density of anisotropy energy, V is the volume of the particle, and H Q () is the critical field at which the energy barrier becomes zero.For ϭ0°, this field equals the anisotropy field which was estimated 8 to be around 100 kOe.The exponent x()ϭ0.86ϩ1.14H Q ()/H Q (0) tends to 2 as goes to zero.At the low-frequency limit 1 (T,H)Ӷ1 the susceptibility given by Eq. ͑1͒ is real and reaches the thermal equilibrium value 0 , usually called the isothermal susceptibility.The Mn 12 Ac equilibrium susceptibility can be obtained numerically differentiating the magnetization versus field curve measured above 3 K, i.e., when all the molecules are superparamagnetic for the characteristic dc measuring times.The equilibrium susceptibility so obtained, at Tϭ5 K and ϭ0°, is shown in Fig. 2.
At high enough frequencies 1 ӷ1 the susceptibility given by Eq. ͑1͒ tends to hf .This theoretical limit was experimentally achieved at Tϭ5 K and ϭ0°measuring the susceptibility at the highest accessible frequency, ϭ980 Hz; the results are also represented in Fig. 2. Finally, in the same figure, the Ј(H) measurements for ϭ15 Hz at Tϭ5 K and ϭ0°are shown.They lie between both limits for all the applied fields and approximate the isothermal limit as H increases ͑although this limit in fact decreases with increasing H and becomes very close to hf above 9 kOe͒.
For classical thermally activated relaxation, the field dependence of Ј is determined by the decrease of the anisotropy energy barrier U as H increases ͓cf.Eq. ͑3͒ above͔.The effective relaxation time 1 given by Arrhenius' law is then shorter and, according to Eq. ͑1͒, the system should approach the equilibrium susceptibility value.Although experimental Ј(H) data indeed approach 0 (H) at high fields, the Ј(H) curve also shows two sharp peaks centered around H 0 ϭ0 and H 1 ϭ4.1(1) kOe ͓B 1 ϭH 1 ϩ4M eq ϭ5.1(1) kG͔.We note that B 1 , that is, the internal field which is seen by the molecule, is very similar to the value B 1 ϭ5 kG, where the first jump appears in magnetic hysteresis loops measured at low temperatures. 4,5To amplify the Ј(H) and Љ(H) peaks at H 1 we first determined, at each temperature, the relaxation time 1 (H 1 ) from the inflection point of the Ј() experimental curve.We then measured Љ(H) and Ј(H) in each isotherm at the frequency which verifies 1 Ϸ1.In other words, we synchronized our experimental exciting frequency to the relaxation rate of the molecules at HϭH 1 .In this condition, according to Eq. ͑1͒, dЈ/d( 1 ) has a maximum.Thus, the susceptibility becomes very sensitive to the dependence of the relaxation time on H.If Eq. ͑1͒ is again considered, the origin of the two Ј(H) peaks could be either the existence of narrow maxima in the equilibrium susceptibility, which are not experimentally observed, or a sudden speeding up of the magnetic relaxation near H 0 and H 1 .In order to clarify this point, we estimated, using Eq.͑1͒ to interpolate between the low-frequency 0 (H) and high-frequency hf (H) limits obtained experimentally, the field dependence that Ј( ϭ15 Hz) would follow if the relaxation mechanism at T ϭ5 K was the thermal activation of the magnetic moment over the anisotropy barrier.The parameters 0 and U(H ϭ0) for ϭ0°, which are given in Table I, were substituted in Eqs.͑2͒ and ͑3͒ to obtain (H).This Ј(H) estimation is shown as a continuous line in Fig. 2. From inspection of Fig. 2, it is clear that the calculated susceptibility differs with the experiment.Thus, Eqs.͑1͒-͑3͒ do not predict the existence of a sharp Ј peak at a finite field H 1 ϭ4.1 kOe.It follows that the field dependence of 1 disagrees with the classical expectation for a thermally activated relaxation mechanism.
In order to determine unambiguously that the ac susceptibility peaks are due to a nonclassical field dependence of the relaxation time, we performed ac susceptibility measurements as a function of the frequency in the 0.025 HzϽϽ1000 Hz range.We obtained the relaxation time 1 fitting the Ј() and Љ() experimental curves to Eq. ͑1͒.Typical experimental curves measured at different field values around H 1 ϭ4.1 kOe are shown in Fig. 3.We note first that the fits are reasonably good except at the high-frequency region.This discrepancy indicates that there are other faster relaxation mechanisms for the magnetic moments, as we expected from the ac (T) data ͑see Fig. 1͒.However, it is clear in Fig. 3 that the Љ() maximum appears at a higher frequency for H 1 ϭ4.1 kOe than for Hϭ3.5 and 5 kOe.The field dependence of 1 at Tϭ5 K is shown in Fig. 4. We where ⌬ n can be interpreted as the field range around H n (nϭ0,1) for which the relaxation time increases with the absolute value of HϪH n .B (H) describes the field dependence of the susceptibility far from HϭH n .In order to fit the experimental data to Eq. ͑4͒, B (H) was chosen to be a constant value for the H 0 peak and a second-order polynomial near H 1 .At Tϭ5 K and ϭ0°, we obtain ⌬ 0 ϭ270(3) Oe and ⌬ 1 ϭ351(9) Oe, respectively, ͑see Fig. 5͒.⌬ 1 is larger than ⌬ 0 for all T and .Experimental Ј(H) and Љ(H) curves corresponding to ϭ25°and three different temperatures Tϭ4.3, 5, and 5.5 K are shown in Fig. 6.We note that Ј and Љ also show two peaks near H 0 ϭ0 and H 1 ϭ4.5 kOe; i.e., the field value H 1 does not depend significantly on temperature.On the other hand, ⌬ 1 increases by 24% as the temperatures raises from 4.3 to 5.5 K whereas ⌬ 0 remains approximately constant.
At any value and orientation of the dc applied magnetic field, the relaxation time strongly increases as the temperature decreases.In fact, the temperature dependence of 1 can be fitted to Arrhenius' law as shown in Fig. 7 for ϭ25°.The values of the activation energy U eff and time constant 0 parameters obtained from this fit of 1 are given in Table I.It is interesting to note that, in agreement with previously reported data, 8 the order of magnitude of the microscopic time 0 does not change with the applied field.We note that for ϭ25°, U is smaller at H 1 ϭ4.5 kOe, which corresponds to the second peak in Fig. 6, than the value obtained for a slightly larger field value Hϭ5 kOe; i.e., the value of U eff is smaller in-tune condition than off-tune condition.Moreover, the relaxation time 1 and the activation energy U obtained at H 0 ϭ0 ͑for ϭ0°and ϭ25°) are larger than the same parameters measured at HϭH 1 () ͑H 1 ϭ4.1 kOe for ϭ0°and H 1 ϭ4.5 kOe for ϭ25°).The observation that U(H) is smaller at H n , with nϭ0,1, than at fields that are off-tune condition is crucial for this work since it points towards the physical origin of the relaxation time minima at 1 (H n ), as we discuss in the next section.Again, this experimental result disagrees with the decrease of U(H) as H increases predicted for classical thermally activated relaxation ͓Eq.͑3͔͒.
In order to investigate the effect of the transverse component of the applied field H x on the magnetic relaxation of Mn 12 Ac, we repeated the same set of ac susceptibility experiments for 0. The Ј(H) and Љ(H) experimental curves show two maxima at HϭH 0 and HϭH 1 for all values ͑see Fig. 8͒.The first Ј peak is always centered about H 0 ϭ0 whereas H 1 () increases as increases.It is very relevant to note that the field value H 1 () at which the first peak appears is such that its component parallel to the z axis is nearly independent on for р60°and approximately satisfies (H 1 ) z ϭH 1 cosϷ4.1 kOe ͑see Figs. 8 and 9͒.For larger angles the width of the Ј(H) peak is too large to define the position of the critical field with enough precision as to assure this statement.In spite of this experimental shortcoming, we infer that the existence of a sharp minimum in 1 (H) at HϭH 1 () is determined by the value of the (H 1 ) z rather than by the total applied field H 1 ().It is also evident from the data exhibited in Fig. 8 that the Ј and Љ maxima become broader and lower as departs from zero.The relaxation time measured under the condition H z ϭ(H 1 ) z fixed, which is shown in Fig. 10, decreases as H x increases, in contrast with low-temperature magnetic relaxation experiments which detected no dependence of 1 on . 16This discrepancy can be due to the very different experimental conditions under which both series of experiments were performed: Tϭ200 mK and variable H z in the experiment of Paulsen and Park Tϭ5 K and H z ϭconst in ours.
When the applied field was parallel to the z axis of the sample, we observed a third peak of Ј near H 2 ϭ8.4 kOe (B 2 ϭ9.7 kG).This peak is shown in the inset of Fig. 8.The value B 2 is not far from the magnetic-field value BϷ10 kG where a jump is observed in the hysteresis loop. 4,5he possibility of observing peaks at multiple values of H 1 higher than two is hindered by the merging, above H ϭ9 kOe, of the isothermal and high-frequency limits of the susceptibility.In Ref. 8, a deep minimum of the relaxation time about Hϭ0 and a second one at Hϭ3 kOe, quite smaller than our H 1 ϭ4.1 kOe value, were observed.The authors suggested the existence of a fine structure of maxima  and minima below 5 kOe.From our data ͑cf.Figs. 4 and 8͒ it seems that the relaxation time minima appear only at H z Ϸn H 1 (0), where n is an integer.

IV. DISCUSSION
Our main experimental results are ͑1͒ When the field is applied parallel to the anisotropy axis of the sample, we observe sharp susceptibility maxima at Hϭ0, 4.1, and 8.4 kOe, which we denominate H n , with nϭ0, 1, and 2, respectively.5][6] This is clear evidence for the existence of a common underlying mechanism.͑2͒ The relaxation time shows minima at these field values.͑3͒ Despite its rather striking field dependence, the relaxation time decreases as T increases, and follows Arrhenius' law for all fields.͑4͒ The corresponding activation energy U eff depends on H; it is lower at the tuning field values HϭH n than for off-tune fields.͑5͒ The tuning condition only depends on the component of the field along the anisotropy axis, and is fulfilled at least when (H 1 ) z ϭH 1 cos Ϸ4.1 kOe.͑6͒ At this tuning field value we have verified that the relaxation time becomes shorter as the transverse component H x increases.͑6͒ The relaxation time at H n ϭnH 1 , decreases as n increases, at least for nϭ0, 1, and 2.
The following spin Hamiltonian has been proposed for this system: 18,19 HϭϪDS z 2 ϩg B S z HϩHЈ ͑5͒ where D is the anisotropy energy constant.The values D Ϸ0.73 K and gϷ1.9, determined by means of magnetization and high-field ESR experiments, will be used here. 17The first term is an uniaxial anisotropy energy, the second term corresponds to the interaction of the magnetic moment with the z component of the applied field, and HЈ is a perturbation which does not commute with S z .We treat the molecule as a Sϭ10 object since, at low temperatures, the population of the Sϭ9 excited multiplet is expected to be small.The eigenstates ͉m͘ of S z are also eigenstates of the unperturbed Hamiltonian.The unperturbed ground state, in the absence of an applied magnetic field, is the ͉Ϯ10͘ doublet, and all ͉Ϯm͘ states are degenerate.The perturbation HЈ induces tunneling between states of opposite spin orientation.The Zeeman term g B S z H z breaks the zero-field degeneracy of the anisotropy term.It follows that one of the two anisotropy energy wells, in which mϾ0 for H z Ͼ0, becomes metastable ͑see Fig. 11͒.For increasing field values the ͉ϩm͘ state increases, while the ͉Ϫm͘ decreases in energy and both cross with the adjacent states.According to Eq. ͑5͒, all unperturbed energy levels on both sides of the energy barrier match at H z ϭnH 1 (0), where H 1 (0)ϭD/g B ͑see Fig. 12͒ and n is an integer.Using the values of D and g given above, we obtain H 1 (0)ϭ5.7 kOe, larger than the observed value by 37%, approximately.The origin of this large discrepancy can be the difference between the externally applied field H and the magnetic field BϭHϩ4M which really interacts with the magnetic moment of the molecule.Therefore, the difference between the applied field H 1 and the corresponding internal field B 1 is expected to be larger if the magnetization of the sample attains its equilibrium value, as in our ac experiments.The theoretical value B 1 (0) ϭ5.7 kG is only about 10% larger than the crossing field B 1 (0)ϭ5.1 kG obtained from either ac or dc experiments. 4,5n our experiment, we apply an oscillating field parallel to the dc field.Therefore, the observed time evolution of the magnetization is induced by the oscillating applied field.During this time evolution, the magnetization approaches its equilibrium value and, consequently, a continuous redistribution of the spin states population follows.Several possible mechanisms have been proposed to explain this evolution for Mn 12 Ac molecules.Below we compare them to our experimental results.
At high temperature, the relaxation time measured experimentally follows Arrhenius' law ͑see Fig. 7͒.It has been proposed 18 that the molecule follows a series of thermally activated Orbach processes to overcome the anisotropy energy barrier U.In each Orbach process, the molecule changes from an initial ͉m͘ state to a final ͉mϮ1͘ state.U is defined as the distance from the bottom of the metastable energy well to the top of the barrier ͑see Fig. 11͒.At zero applied field, Uϭ100DϷ73 K which is the energy difference between the unperturbed states ͉mϭ0͘ and ͉mϭϮ10͘.As H z increases, the energy of one of the wells decreases with respect to the other ͑see Fig. 12͒ and, as a result, the barrier U decreases according to Eq. ͑3͒.Consequently, a monotonic decrease of 1 as H z increases follows in this model ͓see Eq. ͑2͔͒, in contradiction with the two experimentally observed dips exhibited in Fig. 4.
A magnetization quantum tunneling relaxation mechanism has been proposed recently by Politi et al. 19 for the Mn 12 Ac molecules.The unperturbed Hamiltonian given by Eq. ͑5͒ commutes with S z for ϭ0 and quantum transitions between different orientations of the magnetic moment along the easy axis are then forbidden.The perturbation HЈ proposed in Ref. 18 follows from a fourth-order distortion of the uniaxial anisotropy; it induces tunneling between the ͉mϭϪ10͘ unperturbed state, which is assumed to be the initial state of the molecule, and a final ͉ϩm͘ unperturbed state.However, this term only allows tunneling between unperturbed spin states ͉mЈ͘ and ͉mЉ͘ which satisfy mЈϪmЉ ϭ4N, where N is an integer.From the field dependence of the unperturbed Hamiltonian eigenvalues ͑see Fig. 12͒, crossing of such levels would occur only at fields H n with nϭeven.In contradiction with this prediction, we observe susceptibility maxima at all the integer multiples of H 1 (n ϭ0,1,2).The prediction of Ref. 8 that the relaxation time decreases monotonically as H z increases is in marked contrast with our observation of minima at all HϭH n .
From the above discussion we may conclude that the relaxation mechanism underlying our experimental results performed on Mn 12 Ac molecules appears to be neither a classical thermally activated process nor tunneling through the lowest-lying states.We propose a relaxation mechanism which involves three steps: ͑1͒ thermal population of an excited state ͉ϩmЈ͘ from the initial ͉ϩ10͘ unperturbed state, ͑2͒ tunneling through the anisotropy barrier through some excited levels, and ͑3͒ decay to the ground state ͉Ϫ10͘.This process is represented schematically in Fig. 11.Then, the maxima observed in Ј(H) at H z ϭ0, 4.1, and 8.4 kOe would be related to the existence of resonant tunneling between nearly degenerate unperturbed states at these fields, which leads to local minima of the relaxation time 1 ͓see Fig. ͑4͔͒.
The tunneling effects we have observed require thermal activation.This is so because tunneling through the lowestlying energy states is suppressed by fields as small as the ac field amplitude h 0 (ϳ4 Oe) we have applied in our experiments.The corresponding Zeeman energy is much larger than the relevant energy splittings, ⌬E T , that one expects to be generated by HЈ. [19][20][21] Detuning therefore takes place even if the dc field satisfies H z ϭnH 1 .However, ⌬E T increases sharply ͓exponentially fast in ͉m͉ ͑Ref.20͔͒ as one moves up the energy barrier.It is therefore not surprising to find a pair of states ͉m͘ and ͉Ϫmϩn͘, for ͉m͉ sufficiently small, for which ⌬E T ϳg B h 0 is fulfilled.Now, consider temperatures that are not too low, such that the dominant mechanism off resonance ͑that is, for H z away from nH 1 ͒ is thermal activation over the energy barrier.Then tunneling through states ͉m͘ and ͉mЈϭϪmϩn͘ that lie below the energy barrier is to be expected when H z ϭnH 1 , n ϭ0,1,2,... .The relaxation time follows an Arrhenius law with an effective activation energy U eff , which is the energy difference between the metastable state ͉ϩ10͘ and the energy of the tunneling states.We infer this from our experimental results, independently of whether tunneling is phonon assisted or not.
It is worth pointing out that small stray external fields ͑Earth's magnetic field, for instance͒ would not suppress tunneling between these excited states because their tunneling splitting ⌬E T is large ͑see Ref. 20͒.The inhibiting field H D large enough to switch tunneling off is H D Ϸ⌬E T / g B (mϪmЈ).If ͉HϪH n ͉ϾH D for all levels below the energy barrier top, the magnetic moments relax by thermal activation over the classical energy barrier U. Since UϾU eff , ''Zeeman detuning'' leads to the increase of the relaxation time that we have observed experimentally.The interaction of the molecule with a thermal bath as well as hyperfine field effects ͓H hyp Ϸ250 Oe ͑Ref.22͔͒ broaden the unperturbed energy levels.In addition, the local field which is seen by the magnetic moment fluctuates due to thermal modulation of the dipolar interaction with neighboring molecules, leading to homogeneous broadening of the susceptibility peaks.All these effects enlarge the value of H D and, consequently, the experimental halfwidth ⌬ n .The susceptibility peaks are also broadened ͑about 30 Oe͒ by any slight misalignment of the crystallites' c axis from the z axis of the sample.
In the following discussion, we try to find the most effective relaxation channel.The relaxation time 1 is expected to follow approximately Arrhenius' law for thermally activated tunneling ͑for ͉HϪH n ͉ϽH D ͒ as well as for classical overbarrier ͑for ͉HϪH n ͉ϾH D ͒ relaxation.Moreover, the effective energy barrier U eff for tunneling through excited states is smaller than the classical value U for thermally activated overbarrier transitions.At the H 0 ϭ0 in-tune condition, the difference between the calculated classical barrier height U ϭ73 K and the measured value U eff ϭ61 K, scaled by D yields ⌬U/Dϭ16.4,very close to the value ⌬U/Dϭ16 of the ͉mϭϩ4͘ and ͉mϭϪ4͘ tunneling states ͑Fig.12͒.At H 1 ϭ4.1 kOe and H x ϭ0, the calculation of the barrier height yields Uϭ66 K, while the measurement yields U eff ϭ57 K, thus the difference is ⌬U/Dϭ12.3,close to the value ⌬U/Dϭ11 which corresponds to the ͉mϭϪ3͘, ͉mϭϩ4͘ tunneling doublet.We can conclude from these estimations that in the temperature region we are exploring, the doublets involved in tunneling lie about UϪU eff Ϸ9-12 K below the top of the barrier.This estimation is corroborated by the measurements performed at ϭ25°, where the values U eff ϭ51 K for the in-tune field H 1 ϭ4.5 kOe, and U eff ϭ60 K for the off-tune field Hϭ5 kOe were obtained.That is, the difference ⌬Uϭ9 K is identical to the estimated difference at ϭ0.
From the values of 0 obtained from the Arrhenius law fits of the data for 1 obtained at fixed field and varying temperature, collected in Table I, we can conclude that they do not vary too strongly for the different applied fields.If, consequently, we assume the approximation that 0 may be considered as constant we may obtain the field dependence of U eff on H z from the 1 (Tϭ5 K,H z ) data plotted in Fig. 4 using the expression U eff (H z )ϭkT ln͓ 1 (H z )/ 0 ͔.With 0 ϭ6ϫ10 Ϫ8 s, the value for H z ϭ0, the resulting Consider the parallel and transverse components, H z and H x , respectively, of the field that is applied at an angle from the anisotropy axis.We now discuss the observed dependence of the relaxation time as a function of H x , keeping H z ϭ(H 1 ) z ϭconst.The magnetic interaction term, g B H x S x , has two effects: the classical barrier height U(H z ,H x ) decreases as H x increases, and energy levels as well as energy splittings change with H x .Now, which is the most effective channel for tunneling depends on how the Boltzmann factor and the energy splitting change from one energy level to the next one.Furthermore, energy-level spacings at a given energy below the barrier top depend weakly on H x , while the energy splitting of a level with energy E n depends exponentially on UϪE n , but more weakly on H x . 20Consequently, the energy difference between the doublet or doublets which contribute most to magnetic tunneling at H z ϭH 1 and the lower state of the metastable well is roughly given by U eff ϭU(H x )Ϫconst.As mentioned above the most effective tunneling doublet for H z ϭ(H 1 ) z ϭ4.1 kOe and H x ϭ0 is about 9 K below the barrier, for temperatures near 5 K.We therefore estimate the value of 1 versus the transverse component H x at H z ϭH 1 with Arrhenius' law with 0 ϭ1.1ϫ10Ϫ7 s, the value obtained from the fit of the 1 (T) data measured at ϭ0°.The resulting 1 (H x ) curve, shown in Fig. 10, fits the experimental data points reasonably close.Thus, the data we have ob-tained with fields applied at a nonzero angle to the anisotropy ͑i.e., for H x 0͒ axis fit well with the tunneling process that we infer above for H x ϭ0.
Attempts to explain the observed effects by processes taking place near the top of the barrier fail for the reasons that follow.All unperturbed energy levels lie below the top of the barrier for any field H applied along the z direction.These levels lie on a pattern that is repeated when H changes by twice the value of H 1 .If such a process would induce resonances in 1 versus H, they would be spaced at twice the observed amount.Moreover, if the process involves the first one or two levels below the top of the barrier the difference in activation energy ⌬U would amount to an order of magnitude less than observed.

V. CONCLUSIONS
We have performed a detailed ac susceptibility study of the magnetic relaxation of Mn 12 Ac molecules at temperatures above 1.8 K.A possible relaxation mechanism, the tunneling between thermally excited spin states explains qualitatively the experimental results.When the field is applied parallel to the anisotropy axis of the sample we observe sharp susceptibility maxima at the field values H n , with n ϭ0, 1, and 2 that correspond to the crossing of the Zeeman splitted spin levels.The most striking result is that the relaxation time exhibits minima as a function of magnetic field at the crossing field values.The temperature dependence of the relaxation time follows an Arrhenius law.The corresponding activation energy U depends on H, with a value which is lower at HϭH n than at any other field of similar magnitude but off-tune condition.At any of the crossing fields, pairs of levels with opposite spin orientation are tuned in energy so that a non-negligible tunneling probability exists for pairs lying not far below the top of the barrier.The pair with the largest value of tunneling rate times the relevant Boltzmann factor dominates the relaxation process and determines an effective activation energy U eff of the Arrhenius law.This energy is lower than the classical overbarrier flipping process activation energy U, determined at off-tuning field condition.From the difference between both activation energies the level pairs involved in the tunneling have been deduced.
A transverse field hardly modifies the crossing field value (H 1 ) z , while it does reduce the relaxation time.This reduction can be explained as due to a lowering of the effective activation energy of the tunneling process due to the decrease in the height of the barrier caused by the transverse field.
We think that the main questions left to be answered are, ͑1͒ the nature of the perturbing Hamiltonian HЈ, even in the absence of any applied field, ͑2͒ how tunneling takes place, and ͑3͒ how to predict what energy levels contribute most to relaxation as a function of temperature.We believe that the perturbing Hamiltonian HЈ must be linear in spins ͑such as random dipolar or applied magnetic fields, or hyperfine in-teraction͒ but not of fourth order in transverse spins, as proposed by Politi et al., 19 since then the peaks at HϭnH 1 , with nϭodd, would not be observed.
Note added in proof.Magnetization measurements performed on a single crystal have corroborated the presence of hysteresis jumps.

FIG. 1 .
FIG. 1. Magnetic ac susceptibility measured at ϭ5 Hz along the z axis of the sample; ᭹, real component; ᭺, imaginary component.

FIG. 4 .
FIG. 4. Field dependence of the relaxation time for Tϭ5 K and ϭ0°.

FIG. 8 .
FIG. 8. Magnetic ac susceptibility isotherms measured at different orientations , plotted as a function of the component of the applied field H z which is parallel to the easy axis of the sample; ᭹, real component; ᭺, imaginary component.Inset: Enlarged view of the H 2 peak measured at ϭ0°, Tϭ5 K, and ϭ15 Hz.

FIG. 9 .
FIG. 9. Angular dependence of the crossing field component parallel to the easy axis (H 1 ) z .The continuous line represents the condition (H 1 ) z ϭ4.1 kOe.

FIG. 11 .
FIG.11.Energy levels scheme of the unperturbed Hamiltonian HϪHЈ ͓see Eq. ͑5͔͒ at H z ϭH 1 (0).The tunneling process proposed in the text is shown schematically: ͑1͒ Thermal activation from the initial state ͉mϭ10͘ to the excited state ͉mϭ4͘, ͑2͒ tunneling to ͉mϭϪ3͘, and ͑3͒ decay to the ground state ͉mϭϪ10͘.U and U eff are the energy barriers for classical thermally activated relaxation and tunneling between excited states, respectively.

U
eff (H z )/D values show sharp minima at the tuning fields and maxima corresponding to overbarrier process ͑Fig.13͒.In the same figure we have plotted the values of U eff /D obtained from the Arrhenius law fits of the data taken at fixed field and varying temperature ͑Table I͒, observing a reasonable agreement, a proof of the soundness of the 0 ϭconstant approximation.The field dependence of the activation energy U(H z ) for classical overbarrier process has also been plotted in the same figure ͑scaled to D and H 1 ͒.This line coincides nicely with the U eff (H z )/D points at the off-tune field values H z /H 1 ϭ0.6 and at H z /H 1 ϭ1.5.Thus the maximum values decrease as H z increases because the barrier height decreases, and the dominant process for these off-tune field values is overbarrier hopping.

TABLE I .
Activation energy U eff , and prefactor 0 , Arrhenius law ͓Eq.͑2͔͒ parameters obtained from the fit of 1 (T) data, for different strengths and orientations of the applied field.͑Uϭ100, Dϭ73 K͒.
emphasize that 1 does not have a monotonic decrease as H increases but shows minima at H 0 ϭ0 and H 1 ϭ4.1 kOe, in contrast with the classical expectation for a pure thermally activated relaxation process.The Ј(H) experimental data can be fitted near H 0 ϭ0 and H 1 ϭ4.1 kOe to a Lorentzian curve ͑Fig.5͒: