Spin decoherence processes in the S = 1/2 scalene triangular cluster (Cu₃(OH))

In order to estimate the exchange coupling constants, we made spin dimer analysis. A spin exchange parameter J can be written as $J={{J}_{{\rm F}}}+{{J}_{{\rm AF}}}$, where ${{J}_{{\rm F}}}(\gt 0)$ is the ferromagnetic component and ${{J}_{{\rm AF}}}(\lt 0)$ is the antiferromagnetic component. For a spin dimer in which each spin site contains unpaired spin, J_AF is expressed as [33, 34]

$$\begin{eqnarray}&&{{J}_{{\rm AF}}}\approx -\frac{{{(\Delta \varepsilon )}^{2}}}{{{U}_{{\rm eff}}}}\end{eqnarray} \tag{ 1 }$$

where U_eff is the effective on-site repulsion, which is essentially constant for a given compound. If the two spin sites are equivalent, $\Delta \varepsilon$ is the energy difference $\Delta E$ between the two magnetic orbitals representing the spin dimer. When the two spin sites are nonequivalent, ${{(\Delta \varepsilon )}^{2}}={{(\Delta E)}^{2}}-{{(\Delta {{E}^{0}})}^{2}}$, where $\Delta {{E}^{0}}$ is the energy difference between the magnetic orbitals representing each spin site of the spin dimer ($\Delta {{E}^{0}}=0$ if the two spin sites are equivalent).

In our case, the $\Delta E$ and $\Delta {{E}^{0}}$ values for various spin dimers are evaluated by performing extended Hückel tight binding (EHTB) calculations[35]. For a variety of magnetic solids of transition-metal ions, it has been found that their magnetic properties are well described by the ${{(\Delta \varepsilon )}^{2}}$ values obtained from EHTB calculations [36], when both the d orbitals of the transition metal and s/p orbitals of its surrounding ligands are represented by double-ζ–Slater-type orbitals (DZ–STO) [37]. Our calculations were carried out using the atomic parameters summarized in table 1. A radial part of the DZ–STO is expressed as

$$\begin{eqnarray}&&{{r}^{n-1}}[{{c}_{1}}{\rm exp} (-{{\zeta }_{1}}r)+{{c}_{2}}{\rm exp} (-{{\zeta }_{2}}r)],\end{eqnarray} \tag{ 2 }$$

where n is the principal quantum number and the exponents ${{\zeta }_{1}}$ and ${{\zeta }_{2}}$ describe contracted and diffuse STOs, respectively (i.e., ${{\zeta }_{1}}\gt {{\zeta }_{2}}$). The ${{(\Delta \varepsilon )}^{2}}$ values are a sensitive function of the exponent ${{\zeta }_{2}}$ of the diffuse O 2p orbital. The ${{\zeta }_{2}}$ values taken from the results of electronic structure calculations for neutral atoms may not be contracted enough to describe O²⁻ ions [37]. To make the O 2p orbital more contracted, the ${{\zeta }_{2}}$ value should be increased. To quantify how the contraction of the O 2p orbital affects the relative strengths of the spin exchange interactions, we replace ${{\zeta }_{2}}$ with $(1+x){{\zeta }_{2}}$ and calculate the ${{(\Delta \varepsilon )}^{2}}$ values for $x=0.00,0.025,{\rm and}\;0.05$.

Table 1.  List of exponents ${{\zeta }_{i}}$ (i = 1, 2) and valence shell ionization potentials H_ii of Slater-type orbitals ${{\chi }_{i}}$ used for the extended Hückel tight-binding calculation.^a

Atom	${{\chi }_{i}}$	Hii (eV)	${{\zeta }_{1}}$	C1	${{\zeta }_{2}}$	C2
Cu	4s	−11.4	2.151	1.0
Cu	4p	−6.06	1.370	1.0
Cu	3d	−14.0	7.025	0.4473	3.004	0.6978
O	2s	−32.3	2.688	0.7076	1.675	0.3745
O	2p	−14.8	3.694	0.3322	1.659	0.7448
N	2s	−26.0	2.261	0.7297	1.425	0.3455
N	2p	−13.4	3.249	0.2881	1.499	0.7783

^aH_ii are the diagonal matrix elements $\langle i|{{H}_{{\rm eff}}}|i\rangle$, where H_eff is the effective Hamiltonian. In our calculations of the off-diagonal matrix elements ${{H}_{ij}}=\langle i|{{H}_{{\rm eff}}}|j\rangle$, the weighted formula was used. See [38].

The relative values of ${{(\Delta \varepsilon )}^{2}}$ for the superexchange (SE) paths J₁, J₂, and J₃ are summarized in table 2. The strongest SE interaction takes place through J₂, while the weakest SE interaction occurs through J₃. J₁ lies between J₂ and J₃ for all x. These results are compatible with the fact that the bond angle Cu1–O1–Cu3 of J₂ is slightly larger than that Cu1–O1–Cu2 of J₁, i.e. 119.2° versus 111.4°, respectively. In addition, J₃ is mediated by two exchange paths Cu2–O1–Cu3 and Cu2–O2–Cu3 but their bond angles of 95.8 and 101.1° are smaller than those of J₁ and J₂. The relative strengths of the spin exchange interactions obtained from the spin dimer analysis give a reasonable description of magnetic behaviors, which will be discussed in the following section.

Figure 2(a) shows $\chi (T)T$ versus T measured in the temperature range of T = 2–220 K at an external field of ${{\mu }_{0}}H=0.1$ T. $\chi (T)T$ is weakly temperature dependent in the temperature range of 120–220 K and then decreases rapidly for temperatures below 120 K. The nearly constant $\chi (T)T$ at higher temperatures is associated with the mixture of both ${{S}^{T}}=1/2$ and ${{S}^{T}}=3/2$ spin states while the decrease of $\chi (T)T$ at lower temperatures is due to the predominant occupation of the ${{S}^{T}}=1/2$ state over the ${{S}^{T}}=3/2$ one. We find no difference between field-cooling and zero-field-cooling data. This is consistent with a ${{S}^{T}}=1/2$ ground state.

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Since $S=1/2$ Cu²⁺ ions in the scalene triangle (Cu₃(OH)) are coupled by Heisenberg exchange interactions, the minimal spin Hamiltonian in an external field can be written as

$$\begin{eqnarray}&&\hat{H}={{J}_{1}}{{{\bf S}}_{1}}\cdot {{{\bf S}}_{2}}+{{J}_{2}}{{{\bf S}}_{2}}\cdot {{{\bf S}}_{3}}+{{J}_{3}}{{{\bf S}}_{3}}\cdot {{{\bf S}}_{1}}-g{{\mu }_{B}}{\bf H}\cdot \mathop{\sum }\limits_{i=1}^{3}{{{\bf S}}_{i}},\end{eqnarray} \tag{ 3 }$$

where g is the average g-factor, ${{\mu }_{B}}$ is the Bohr magneton, J_i is the exchange coupling constant, and ${{{\bf S}}_{i}}$ is the spin operator. For a realistic Hamiltonian, Dzyaloshinskii–Moriya (DM) interactions and a site-dependent g-tensor should be added to equation (3) as (Cu₃(OH)) has the structural distortions from an equilateral triangle. We note that the Hamiltonian (3) cannot open anti-crossing energy gaps between the ${{S}^{T}}=1/2$ and the ${{S}^{T}}=3/2$ state. As a consequence, the magnetization jumps in a magnetization curve cannot be described strictly within this Hamiltonian (see figure 2(b)). Since the level crossings occur around 50 T, however, it is practically impossible to determine uniquely all possible magnetic parameters including DM interactions. Therefore, we will proceed with the Hamiltonian (3) for the sake of simplicity.

For the calculation of the equilibrium magnetization, we used the MAGPACK software, which employs an irreducible tensor operator technique [39]. We obtain a satisfactory agreement between the theoretical and the magnetic susceptibility data with the fitting parameters ${{J}_{1}}=-43.5$ K, ${{J}_{2}}=-53.0$ K, and ${{J}_{3}}=-37.7$ K, and g = 2.16 (see the solid line in figure 2(a)). We note that the determined g-factor corresponds to an average of Cu²⁺ ions (see section 3.3). In addition, the magnetic parameters are in line with the ratios of J_i extracted from the spin dimer analysis as listed in table 2.

Based on the obtained fitting parameters, we determine the energy level diagram versus an applied magnetic field as plotted in figure 2(c). The two degenerated ${{S}^{T}}=1/2$ states are lifted due to strong distortions to an scalene triangle. The splitting energy between doublets amounts to about 10 K, which corresponds to an average of the energy difference between J_i, that is, $|{{J}_{i}}-{{J}_{j}}|$ = 5.8–15.3 K. The energy separation between the ${{S}^{T}}=1/2$ doublet and ${{S}^{T}}=3/2$ quartet state is given by about 70 K. The ${{S}^{T}}=1/2$ state crosses successively with the ${{S}^{T}}=3/2$ state at ${{\mu }_{0}}{{H}_{c1}}=48$ and ${{\mu }_{0}}{{H}_{c2}}=56$ T. This induces a field-induced switching of the ground state to the ${{S}^{T}}=3/2$ state, leading to the $2{{\mu }_{B}}$ magnetization jump. In figure 2(b) we plot the pulsed field magnetization curve (violet lines) along with the static field magnetization one (green open circles).

To crosscheck the validity of the determined magnetic parameters, we simulate the pulsed field magnetization curve as detailed below. In doing that, we introduce the anti-crossing energy gap at ${{\mu }_{0}}{{H}_{c}}=56$ T (see equation (7)), which is absent in equation (3). The resulting ground state is shown as the dashed line in figure 2(c). This procedure is justified because the distorted triangle cluster usually hosts the DM interaction, allowing an avoided level crossing gap [15, 23, 24]. In contrast to the magnetization below 15 T, the pulsed field magnetization above 15 T lacks hysteresis. Thus, the equilibrium magnetization process may suffice to describe the high-field magnetic behavior. The equilibrium magnetization is calculated with a usual thermodynamic average. Due to a large energy separation between the ground and the first excited states only the ground state is relevant. The nice agreement between the calculated and the experimental curve is found for fields up to 40 T (see the red line in figure 2(b)). The discrepancy seen for fields above 40 T may be due to the presence of another avoided level crossing at 48 T in the upper doublet state, which is not considered in the equilibrium magnetization simulation. The magnetization process below 15 T will be discussed below.

We measured hysteresis loop with the maximum pulse field of ${{\mu }_{0}}{{H}_{{\rm max} }}=16$ T. As shown in figure 3(a), the detailed feature of M(H) relies on the time structure of a pulse field (see the inset of figure 3(b)). In the up sweep, the pulsed field magnetization is smaller than the equilibrium one while in the down sweep it becomes bigger. This means that a spin temperature is significantly higher (lower) than a bath temperature in the up (down) sweep. We take the derivative of the magnetization, ${\rm d}M(H)/{\rm d}H$ to detail the magnetization structure. The results are plotted in figure 3(b). We can identify a sharp peak at a zero field, suggesting the presence of a zero-field gap. When a field sweep is sufficiently slow, the transition probability between the ${{S}^{T}}=1/2$ levels is determined by the Landau–Zener–Stückelberg (LZS) model and their populations equilibrate with the Boltzmann distribution through spin–phonon transitions. In a fast field sweep, a majority of spins in the higher ${{S}^{T}}=1/2$ levels are out of equilibrium because the energy flow from the lattice to the spins is not sufficient to reach equilibrium. Indeed, the field sweep rate of $4-8\times {{10}^{-4}}$ T s⁻¹ is comparable to the electron spin–lattice relaxation time of ${{T}_{1}}\sim {{10}^{-4}}$s (vide infra). As a consequence, the spin temperature is higher than the cryostat temperature. In the down sweep, conversely, the field variation is too fast for the spins in the lower ${{S}^{T}}=1/2$ levels to populate to the higher levels.

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To describe this behavior quantitatively, we simulate numerically the magnetization hysteresis based on the Bloch-type master equation [40, 41]

$$\begin{eqnarray}&&\frac{{\rm d}}{{\rm d}t}{{M}_{d}}(t,\theta )=\frac{1}{\tau (T,B(t),\theta )}[{{M}_{{\rm eq}}}(T,B(t),\theta )-{{M}_{d}}(t,\theta )],\end{eqnarray} \tag{ 4 }$$

where τ is the relaxation time and M_eq(M_d) is the equilibrium (dynamic) magnetization as a function of time and angle θ. The relaxation rate comprises three terms $1/\tau =1/{{\tau }_{{\rm thermal}}}+1/{{\tau }_{{\rm LZS}}}+1/{{\tau }_{{\rm res}}}$: (i) thermal relaxation $1/{{\tau }_{{\rm thermal}}}$, (ii) LZS transition $1/{{\tau }_{{\rm LZS}}}$, and (iii) residual relaxation $1/{{\tau }_{{\rm res}}}$. $1/{{\tau }_{{\rm res}}}$ represents the relaxation processes other than the thermal relaxation.

Among different thermal relaxation processes, we consider a single-phonon relaxation process given as [42]

$$\begin{eqnarray}\begin{array}{rcl} 1/{{\tau }_{{\rm thermal}}} & = & \frac{3{{g}^{3}}\mu _{B}^{3}V_{{\rm sp}}^{2}}{2\pi \rho {{v}^{5}}{{\hbar }^{4}}}B{{(t)}^{3}}{\rm coth} \left[ \frac{g{{\mu }_{B}}B(t)}{2{{k}_{B}}T} \right] \\ {} & = & AB{{(t)}^{3}}{\rm coth} \left[ \frac{g{{\mu }_{B}}B(t)}{2{{k}_{B}}T} \right], \\ \end{array}\end{eqnarray} \tag{ 5 }$$

where g is the Lande g-factor, ρ is the mass density, v is the sound velocity, B(t) is the time-varying field strength and V_sp is the characteristic energy modulation of the spin–phonon coupling mechanism.

The LZS transition rate is related to the LZS transition probability P_LZS through $1/{{\tau }_{{\rm LZS}}}=\alpha {{P}_{{\rm LZS}}}$ where α is a proportionality constant. P_LZS is given as [43]

$$\begin{eqnarray}&&{{P}_{{\rm LZS},M\to M^{\prime} }}=1-{\rm exp} \left( -\frac{\pi \Delta _{M,M^{\prime} }^{2}}{2\hbar g{{\mu }_{B}}|M-M^{\prime} |\frac{{\rm d}B}{{\rm d}t}} \right).\end{eqnarray} \tag{ 6 }$$

Here $\Delta _{M,M^{\prime} }^{2}$ is the anti-crossing energy gap between the two states with spin quantum number M and M' expressed as [41]

$$\begin{eqnarray}&&\Delta _{M,M^{\prime} }^{2}=\sqrt{{{({{\Delta }_{{\rm zf}}})}^{2}}+{{[g{{\mu }_{B}}(M-M^{\prime} )B]}^{2}}}-|g{{\mu }_{B}}(M-M^{\prime} )B|,\end{eqnarray} \tag{ 7 }$$

where ${{\Delta }_{{\rm zf}}}$ is the minimum energy gap between unperturbed energy levels.

Best fits of the experimental data were obtained with the parameters: A = 81.0 Hz T⁻³, $1/{{\tau }_{{\rm res}}}=324.0$ Hz, ${{\Delta }_{{\rm zf}}}=5.1$ K and $\alpha =1386$ Hz. The simulated curve is plotted as the red solid lines in figure 3(a). As can be seen, the simulation reproduces reasonably the up sweep of the hysteresis loop. However, there is a noticeable discrepancy between the calculated and experimental magnetization curve in the down sweep. This may be due to the fact that the thermal relaxation rate is not described within the single-phonon relaxation process (see section 3.4). Further, it should be noted that the anti-crossing gap ${{\Delta }_{{\rm zf}}}$ at ${{\mu }_{0}}H=0$ T is much larger than the magnitude of intermolecular dipole and hyperfine interactions. This indicates that the zero-field magnetization jump is linked to the energy splitting between the doublets rather than in the intra-doublets.

Figure 4(a) shows the temperature dependence of the EPR signal recorded at $\nu =240$ GHz. At T = 5 K, a single peak originates from the electron spin transition between the ${{S}^{T}}=1/2$ level, i.e., $|\frac{1}{2};-\frac{1}{2}\rangle \to |\frac{1}{2};\frac{1}{2}\rangle$. With increasing temperature, the signal shifts to lower fields and broadens. We plot the temperature dependences of the peak-to-peak linewidth ($\Delta {{H}_{{\rm pp}}}$) and resonance field (H_res) in figures 4(b) and (c). $\Delta {{H}_{{\rm pp}}}(T)$ [H_res(T)] increases (decreases) in a monotonic manner for temperatures above 30 K. This is attributed to a gradual population of spins to the excited ${{S}^{T}}=3/2$ levels. At high temperatures, the EPR signal is given by a sum of the three resonance lines; $|\frac{3}{2};-\frac{3}{2}\rangle \to |\frac{3}{2};-\frac{1}{2}\rangle$, $|\frac{3}{2};-\frac{1}{2}\rangle \to |\frac{3}{2};\frac{1}{2}\rangle$, and $|\frac{3}{2};\frac{1}{2}\rangle \to |\frac{3}{2};\frac{3}{2}\rangle$. Due to the structural distortions and anisotropies, the energy separation between the higher levels increases slightly. This leads to the shift of H_res(T) to lower fields and the increase of $\Delta {{H}_{{\rm pp}}}$ at elevated temperatures.

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Figure 5 shows the angular dependence of the EPR spectra recorded at T = 5 K by varying from −2° to 179°. The angle is measured between the triangle plane and the external magnetic field. The angular dependence of H_res(T) is described by the standard relation ${{H}_{{\rm res}}}=\sqrt{{{H}_{\bot }}{{{\rm sin} }^{2}}\theta +{{H}_{\parallel }}{{{\rm cos} }^{2}}\theta }$, where ${{H}_{\bot }}$ and ${{H}_{\parallel }}$ is the resonance field perpendicular and parallel to the triangular plane, respectively. By using the relation $g=h\nu /{{\mu }_{B}}{{H}_{{\rm res}}}$, we can determine anisotropy in g-factors, ${{g}_{\bot }}=2.10(4)$ and ${{g}_{\parallel }}=2.21(3)$. The substantial deviation of the g-factor from 2 confirms the structural analysis that shows the axial elongation of square pyramid environments.

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To characterize spin decoherence processes, we performed a 240 GHz pulsed EPR experiment. The spin–lattice relaxation time T₁ was measured by using an inversion-recovery pulse sequence ($\pi -t-\pi /2-\tau -\pi -\tau -{\rm echo}$) with varying t and fixed $\tau =300$ ns. The maximum available power is about 20 mW and the typical $\pi /2$ pulse length is 260 ns. Figure 6(a) shows the echo decay curve recorded at ${{\mu }_{0}}H=7.813\;$ T and T = 1.45 K. It is fitted to a double exponential function, $I=A{\rm exp} (-T/{{T}_{{\rm long}}})+B{\rm exp} (-T/{{T}_{{\rm short}}})$ with ${{T}_{{\rm long}}}=153\;\mu {\rm s}$ and ${{T}_{{\rm short}}}=63\;\mu {\rm s}$. The short T₁ is likely related to a fast spectral diffusion while the long T₁ pertains to the spin–lattice relaxation time. The spectral diffusion can be caused by molecular motion, exchange interactions, nuclear spin flip-flops, or electron–nuclear cross-relaxation. Here we note that the obtained T₁ is shorter than that of (Cu₃–X) by a factor two [31].

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Figure 6(c) plots $1/{{T}_{1}}$ versus $1/T$ in a semi-logarithmic scale. The spin echo signal becomes significantly weak for temperatures above 2.4 K so that T₁ cannot be unambiguously measured for $T\gt 2.4$ K. In general, a spin–lattice relaxation rate is given by a combination of three terms [42]: (i) direct process with $1/{{T}_{1}}\propto {\rm coth} (\hbar \omega /2{{k}_{B}}T)$, (ii) Raman process with $1/{{T}_{1}}\propto {{T}^{3+2\,m}}$ where m is the spectral dimensionality, and (iii) Orbach process with $1/{{T}_{1}}\propto 1/[{\rm exp} ({{\Delta }_{O}}/{{k}_{B}}T)-1]$. We find that our data are well described by an activation form $1/{{T}_{1}}\propto {\rm exp} (-{{\Delta }_{O}}/T)$ with ${{\Delta }_{O}}=7.4(6)$ K. This corresponds to the approximation of the Orbach process when ${{\Delta }_{O}}\gg {{k}_{B}}T$. This suggests that the Orbach process governs the relaxation in a low-temperature regime. We note that the Raman process is negligible as the T₁ calculation for the (V₁₅) cluster shows [44]. In addition, the direct process is temperature independent at low temperatures. The Orbach process involves a transfer of spins between the ground doublet ${{S}^{T}}=1/2$ via phonons. Thus, the relaxation rate relies on the number of phonons with energy ${{\Delta }_{O}}$. Noticeably, the intra-doublet splitting of $\Delta =10.2(3)$ K at ${{\mu }_{0}}H=7.813\;$ T is close to the value of ${{\Delta }_{O}}=7.4(6)$ K. Therefore, we conclude that the Orbach process takes place through the transition between the ${{S}^{T}}=1/2$ states. This is further supported by the failure of the single-phonon relaxation process in describing the dynamic magnetization process discussed above. We note that our result is contrasted with the case of (Cu₃–X) where the spin–lattice relaxation is dominated by a direct process [31]. The absence of the Orbach process in (Cu₃–X) can be ascribed to a small energy scale of J_i, which is one order of magnitude smaller than that of (Cu₃(OH)).

Using the standard Hahn echo pulse sequence ($\pi /2-\tau -\pi -\tau -{\rm echo}$) we measured the variation of a spin–spin relaxation time, T₂ with temperature. Figure 6(b) exhibits decay of the integrated Hahn echo at ${{\mu }_{0}}H=7.813\;$ T (the transition 1 in the inset of figure 6(d)) as a function of the delay time $2\tau$. The echo intensity decay is fitted to a single exponential function $I\propto {\rm exp} (-2\tau /{{T}_{2}})$ with ${{T}_{2}}=264.(7)$ ns at T = 1.45 K.

From extensive theoretical and experimental works [14, 15, 45], it is well known that for molecular magnets, decoherence is solely determined by three environmental sources: (i) phonons, (ii) nuclear spins, and (iii) intermolecular dipolar interactions. The environmental decoherence time for (Fe₈) can be theoretically extended up to about $500\;\mu$s by optimizing temperature and external fields. Under accessible experimental conditions, however, the decoherence time is limited to an order of microseconds: ${{T}_{2}}\sim 0.63\;\mu$s for (Fe₈) [14], $0.34\;\mu$s for (V₁₅) [15], and $0.75\;\mu$s for (Cu₃–X) molecule magnet [31]. The nuclear spins were identified as a main source of decoherence for (V₁₅), in which T₂ was measured at a low field of ${{\mu }_{0}}H=0.336$ T [15]. Takahashi et al [14] have shown that at high fields the nuclear spin decoherence becomes less significant than the dipolar and the phonon decoherence. As T₂ of (Cu₃(OH)) was determined at a high field of ${{\mu }_{0}}H=7.813$ T, we expect an appreciable contribution of a T₁ mechanism to decoherence in case of (Cu₃(OH)) in contrast to (V₁₅).

Similar to the T₁ relaxation time, ${{T}_{2}}\sim 0.26\;\mu$s of (Cu₃(OH)) is also reduced by several factors compared to that of (Cu₃–X). Since both (Cu₃(OH)) and (Cu₃–X) were measured at the same field and contain the same type of nuclear spins, the shortening of T₂ in (Cu₃(OH)) is mainly caused by the Orbach mechanism.

Lastly, we turn to the temperature dependence of $1/{{T}_{2}}$. Under a high external field of ${{\mu }_{0}}H=7.813$ T, most of the spins are polarized to the $|\frac{1}{2};-\frac{1}{2}\rangle$ state and thus a spin flip-flop process is mainly responsible for the T-dependence of $1/{{T}_{2}}$ [46]. This is modeled by a spin bath fluctuation theory [45, 47];

$$\begin{eqnarray}&&\frac{1}{{{T}_{2}}}=A\mathop{\sum }\limits_{{{m}_{S}}=1}^{7}W({{m}_{S}})P({{m}_{S}})P({{m}_{S}}+1)+{{\Gamma }_{{\rm res}}},\end{eqnarray} \tag{ 8 }$$

where A is a temperature independent parameter, ${{\Gamma }_{{\rm res}}}$ a residual relaxation rate, $W({{m}_{S}})$ the flip-flop transition probability for the m_Sth state with $\Delta {{m}_{S}}=\pm 1$, and $P({{m}_{S}})={\rm exp} (-E({{m}_{S}})/{{k}_{B}}T)/Z$, where Z is the partition function. The experimental data agree with the theoretical calculation obtained by equation (8) with the fitting parameters $W(1)=1.2(2)\mu {{{\rm s}}^{-1}}$ and ${{\Gamma }_{{\rm res}}}=4.0(7)\mu {{{\rm s}}^{-1}}$. This result does not necessarily contradict the above conclusion that the phonon process is substantially involved in decoherence. This is because the measured temperature window of T = 1.45–2.4 K is rather narrow to examine an additional contribution from the Orbach process which has the similar functional form and energy gap of ${{\Delta }_{O}}=7.4(6)$ K as the spin bath theory.