# 14 Thermal equilibrium and superparamagnetism

## 14.1 Approach to thermal equilibrium

Let us go back, for simplicity, to the case in which \(\Omega =0\) in Eq.(13.3) and the magnetic field \(B\) is kept constant in time. The corresponding Hamiltonian is diagonal on the basis \(|\uparrow \rangle\) and \(|\downarrow \rangle\). Consistently with Boltzmann statistics, at thermal equilibrium the density matrix should take the following*diagonal*form \[\begin{equation} \rho_{\sigma\sigma '}^{\rm eq} = \begin{pmatrix} \rho_{\uparrow\uparrow}^{\rm eq} & 0 \\ 0& \rho_{\downarrow\downarrow}^{\rm eq} \end{pmatrix} \, \end{equation}\] where \[\begin{equation} \begin{cases} & \rho_{\uparrow\uparrow}^{\rm eq} = \frac{1}{\mathcal{Z}} {\rm e}^{-\beta\hbar \omega_L/2 } \\ & \rho_{\downarrow\downarrow}^{\rm eq} = \frac{1}{\mathcal{Z}} {\rm e}^{\beta\hbar \omega_L/2 } %\exp\left(\frac{1}{2}\beta\hbar \omega_L \right) \end{cases} \, , \end{equation}\] \(\beta = 1/(k_{\rm B} T)\) and the partition function is \[\begin{equation} \mathcal{Z} ={\rm e}^{-\beta\hbar \omega_L/2 } + {\rm e}^{\beta\hbar \omega_L/2 } \,. \end{equation}\] Thermal averages of individual spin components can be obtained using the relation \[\begin{equation} \tag{14.1} \langle \hat{S}^\alpha \rangle_{\rm th} = {\rm Tr} \{\rho(t)\hat{S}^\alpha \} \end{equation}\] with \(\alpha = x,y,z\). The calculation follows the same steps as Eqs.(13.9)–(13.11) and yields \[\begin{equation} \label{th_ave_S_proj_1} \tag{14.2} \begin{cases} &\langle \hat{S}^x \rangle_{\rm th} =0 \\ &\langle \hat{S}^y \rangle_{\rm th} =0 \\ &\langle \hat{S}^z \rangle_{\rm th} = -\frac{1}{2} {\rm tanh}\left(\frac{1}{2}\beta g\mu_{\rm B} B \right) = -\frac{1}{2} {\rm tanh}\left(\frac{1}{2}\beta\hbar \omega_L \right)\,. \end{cases} \end{equation}\] The tanh-like dependence of the \(z\) component on the ratio \(B/T\) is typical of the paramagnetic response for a spin \(S=1/2\), namely it corresponds to the dependence of the Brillouin function for \(S=1/2\) (see later). Equation(14.2) indicates that two instances should occur when thermal equilibrium is reached:

\(\bf a\) – the \(x\) and \(y\) components of spin averages should vanish

\(\bf b\) – the \(z\) component should

*equal*its thermal-equilibrium value.

The point \(\bf a\)) implies the absence of information on the phase of the wave function (also called quantum incoherence condition). The point \(\bf b\)) requires that energy be distributed according to the Boltzmann statistics. From what seen in the previous sections it should be clear that nor Larmor precession nor quantum tunneling (Landau-Zener formula) can drive the system towards the state in which both conditions \(\bf a\)) and \(\bf b\)) are fulfilled. Therefore, the approach to thermal equilibrium is quite generally described introducing two phenomenological relaxation times called \(T_1\) (spin-lattice) and \(T_2\) (spin-spin). We start considering the former which describes the relaxation of the spin projection along the field, \(\langle \hat{S}^z \rangle_{\rm th}\). Consistently with Eq.(13.11), \(\langle \hat{S}^z \rangle\) is proportional to the difference \(\rho_{\uparrow\uparrow}(t) -\rho_{\downarrow\downarrow}(t)\). The problem is then reduced to defining a phenomenological, but plausible,

*stochastic*equation which describes the evolution in time of \(\rho_{\downarrow\downarrow}(t)\) and \(\rho_{\uparrow\uparrow}(t)\). As already said, these terms of the density matrix represent the probabilities of having the spin at time \(t\) in the

*eigenstate*\(|\downarrow\rangle\) or in the

*eigenstate*\(|\uparrow\rangle\), respectively

^{46}, with the normalization condition \(\rho_{\downarrow\downarrow}(t)+\rho_{\uparrow\uparrow}(t) =1\) holding at every time. One can define the following

*master equation*\[\begin{equation} \label{ME} \tag{14.3} \begin{cases} &\dot{\rho}_{\downarrow\downarrow} = \nu_{\uparrow} \rho_{\uparrow\uparrow} - \nu_{\downarrow} \rho_{\downarrow\downarrow} \\ &\dot{\rho}_{\uparrow\uparrow} = \nu_{\downarrow} \rho_{\downarrow\downarrow} - \nu_{\uparrow} \rho_{\uparrow\uparrow} \end{cases} \end{equation}\] with \(\nu_{\uparrow}\) being the transition rate from \(|\uparrow\rangle\) to \(|\downarrow\rangle\) and \(\nu_{\downarrow}\) the reverse rate. These rates result from the coupling of the spin with the environment, identified with a thermal bath with which the spin is allowed to exchange energy. Thus, the specific form of \(\nu_{\uparrow}\) and \(\nu_{\downarrow}\) depends on the details of the coupling between the spin components and the degrees of freedom of the bath.

In particular, since at thermal equilibrium populations should not vary, the *detailed-balance* condition has to be fulfilled:
\[\begin{equation}
\label{detailed-balance}
\tag{14.4}
\frac{\nu_{\uparrow}}{\nu_{\downarrow}} = \frac{\rho_{\downarrow\downarrow}^{\rm eq}}{\rho_{\uparrow\uparrow}^{\rm eq}} = {\rm e}^{\beta \hbar \omega_{\rm L}}\,.
\end{equation}\]
The above condition simply states that the escape rate from the state \(|\uparrow\rangle\), higher in energy, is exponentially larger than the escape rate from the ground state \(|\downarrow\rangle\) (see sketch in Fig.14.1). The detailed balance is a relatively general principle underlying the approach to thermal equilibrium in systems with quantized energy levels. Defining the population difference as \(n= \rho_{\downarrow\downarrow} -\rho_{\uparrow\uparrow}\), the pair of equations(14.3) reduces to
\[\begin{equation}
\label{ME_1}
\tag{14.5}
\dot{n} = (\nu_{\uparrow}-\nu_{\downarrow}) - (\nu_{\uparrow}+\nu_{\downarrow}) n \,,
\end{equation}\]
whose solution is
\[\begin{equation}
\label{n_t}
\tag{14.6}
n(t)= n^{\rm eq} + \left(n^0 - n^{\rm eq}\right) \, {\rm e}^{-t/T_1} \, .
\end{equation}\]
In Eq.(14.6) \(n^0 = \rho_{\downarrow\downarrow}^0 -\rho_{\uparrow\uparrow}^0\) is the difference between the initial occupation probabilities of \(|\downarrow\rangle\) and \(|\uparrow\rangle\), the constant

\[\begin{equation}
\label{T_1}
\tag{14.7}
\frac{1}{T_1}= \nu_{\downarrow} + \nu_{\uparrow}
\end{equation}\]
defines the relaxation time, and
\[\begin{equation}
\label{n_equilibrium}
\tag{14.8}
n^{\rm eq}= \frac{\nu_{\uparrow}-\nu_{\downarrow}}{\nu_{\uparrow}+\nu_{\downarrow}} = \frac{{\rm e}^{ \beta \hbar \omega_L } -1 }{{\rm e}^{ \beta \hbar \omega_L} +1 }
= {\rm tanh}\left(\frac{1}{2}\beta\hbar \omega_L \right)
\end{equation}\]
is the difference between equilibrium populations (in the second passage the detailed-balance condition(14.4) was used). The equilibrium value of the difference of populations \(n^{\rm eq}\) is indeed proportional to the thermal average \(\langle \hat{S}^z \rangle_{\rm th}\), as it is evident from the \({\rm tanh}(\dots)\) dependence typical of the Brillouin function for a spin 1/2 (see also Eq.(14.2)).
Since this relaxation process implies an exchange of energy between the spin system and the crystal lattice, \(T_1\) is called *spin-lattice relaxation time*.
Note that the detailed-balance condition(14.4) does not specify a functional form for the transition rates \(\nu_{\uparrow}\) and \(\nu_{\downarrow}\), and in fact several choices are possible.

### Bloch equation and spin-spin relaxation \(T_2\)

According to Eqs.(14.2), the spin components (\(x\) and \(y\)) transverse to the applied field should relax to zero at equilibrium, with another characteristic relaxation time \(T_2\). Differently from the longitudinal relaxation process, this decay of spin components transverse to the applied field conserves the energy. In fact, the energy of the spin system in a static magnetic field is determined only by the spin projection along \(\vec B\) itself. Therefore, transverse relaxation does not necessarily require an exchange of energy with the reservoir. For this reason \(T_2\) is called *spin-spin relaxation time*. Equation(13.4) can be modified to incorporate both these phenomenological relaxation terms to obtain (for \(\Omega =0\))
\[\begin{equation}
\label{rho_T_1_and_T_2}
\tag{14.9}
\begin{split}
\frac{\partial }{\partial t} \rho_{\uparrow \uparrow} & = -\nu_{\uparrow} \rho_{\uparrow\uparrow} + \nu_{\downarrow} \rho_{\downarrow\downarrow} \\
\frac{\partial }{\partial t} \rho_{\uparrow \downarrow} & = -i \omega_{\rm L} \rho_{\uparrow \downarrow} - T_2^{-1} \rho_{\uparrow \downarrow} \\
\frac{\partial }{\partial t} \rho_{\downarrow\uparrow} & = i \omega_{\rm L} \rho_{\downarrow\uparrow} - T_2^{-1} \rho_{\downarrow\uparrow}\\
\frac{\partial }{\partial t} \rho_{\downarrow \downarrow} & = - \nu_{\downarrow} \rho_{\downarrow\downarrow} + \nu_{\uparrow} \rho_{\uparrow\uparrow} \,.
\end{split}
\end{equation}\]
The density matrix resulting from the integration of this set of equation reads
\[\begin{equation}
\label{rho_matrix_T_1_and_T_2}
\tag{14.10}
\rho_{\sigma,\, \sigma ' } =
\begin{pmatrix}
\cos^2\left(\frac{\theta}{2}\right) -\frac{1}{2}\left(n^0 - n^{\rm eq}\right){\rm e}^{-t/T_1}
& \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right) {\rm e }^{-i\omega_{\rm L} t} {\rm e }^{-t/T_2} \\
\sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right) {\rm e }^{i\omega_{\rm L} t} {\rm e }^{-t/T_2}
& \sin^2\left(\frac{\theta}{2}\right) +\frac{1}{2}\left(n^0 - n^{\rm eq}\right){\rm e}^{-t/T_1}
\end{pmatrix}
\end{equation}\]
where we made use of Eq.(13.14) for the initial amplitudes \(a\) and \(b\).
The density matrix(14.10) is consistent with the following time dependence of averaged spin components
\[\begin{equation}
\label{S_QM_to_thermal}
\tag{14.11}
\begin{cases}
&\langle \hat{S}^x(t) \rangle = S\,\sin\theta \cos(\omega_{\rm L} t) \,{\rm e }^{-t/T_2} \\
&\langle \hat{S}^y(t) \rangle = S\,\sin\theta \sin(\omega_{\rm L} t) \,{\rm e }^{-t/T_2} \\
&\langle \hat{S}^z(t) \rangle = -S\, n(t)
\end{cases}
\end{equation}\]
with \(S=1/2\), in the present case, and
\[\begin{equation}
\label{n_t_1}
\tag{14.12}
n(t)= n^{\rm eq} \,\left( 1- {\rm e}^{-t/T_1} \right) - \cos\theta\, {\rm e}^{-t/T_1} \,,
\end{equation}\]

with \(n^{\rm eq}\) given by Eq.(14.8).

The time dependence described by Eq.(14.11) can be extrapolated to the macroscopic magnetization \(\vec M\) of an *ensemble* of paramagnetic particles to obtain the Bloch equation

\[\begin{equation}
\label{Bloch_Eqs}
\tag{14.13}
\begin{split}
&\dot{M}^x=-\frac{1}{T_2} M^x - \gamma \left( \vec{M} \times \vec{B}\right)^x \\
&\dot{M}^y=-\frac{1}{T_2} M^y - \gamma \left( \vec{M} \times \vec{B}\right)^y \\
&\dot{M}^z=\frac{ M^{z,\rm eq} -M^z}{T_1} - \gamma \left( \vec{M} \times \vec{B}\right)^z
\end{split}
\end{equation}\]
named after Felix Bloch who first proposed it.
This vectorial equation is phenomenological and, as such, has limitations. Nevertheless, it still represents the basic framework to describe magnetic resonance phenomena.

A simple mechanism that determines \(T_2\) of a nucleus in a solid is the dipolar field generated by the magnetic moments of other nuclei. Limiting ourselves to consider nearest neighboring nuclei, we will call this field \(\vec B_{\rm loc}\). If all nuclei were to precess in phase at time \(t=0\), the dipolar field produced by one nucleus on another will sum up to the external, static field, eventually leading to dephasing. As a rough estimate one can assume \(T_2\) to be of the order of \(1/(\gamma B_{\rm loc})\). With some realistic numbers this time scale falls in the range of 100 \(\mu\)s for NMR experiments.

### Stochastic resonance and spin-lattice relation time \(T_1\)

A.c. susceptibility can be used to determine the spin-lattice relaxation time \(T_1\) and its temperature dependence by measuring the magnetic response of a sample to small oscillating fields with standard magnetometers (e.g., SQUID or VSM), \(\omega\) typically falling in the range \(1-10^5\) Hz.The time scales probed in these experiments are much longer than those accessed in magnetic resonance experiments, like ESR or NMR, and the relevant peak in the susceptibility does not occur for \(\omega\simeq \omega_{\rm L}\). Under these conditions, the precessional terms can be dropped from the Bloch equation and some dynamics remains only along the direction of the applied field

\[\begin{equation}
\label{Bloch_Eqs_T_1}
\tag{14.14}
\dot{M}^z=\frac{ M^{z, \rm eq} -M^z}{T_1}
\end{equation}\]
identified with \(z\). For a system that does not possess a magnetically ordered phase at finite temperature, like a spin chain or a SMM or a SIM, the equilibrium magnetization for small applied fields is given by \(M^{z, \rm eq}=\chi^{\rm eq}B^z\). This relation can be extended to a.c. fields for quasi-static conditions, that is with frequency \(\omega\ll \omega_{\rm L}\). Replacing \(M^{z, \rm eq}\) with \(\chi^{\rm eq} B' {\rm e}^{-i \omega t}\) in Eq.(14.14) yields
\[\begin{equation}
\dot{M}^z=\frac{\chi^{\rm eq} B' {\rm e}^{-i \omega t} -M^z}{T_1}
\end{equation}\]
whose stationary solution is
\[\begin{equation}
M^z(t)=\frac{\chi^{\rm eq}}{1-i\omega T_1} B' {\rm e}^{-i \omega t}\,.
\end{equation}\]
The equation above states that, under the hypotheses considered here, the Bloch equation is consistent with an a.c. susceptibility
\[\begin{equation}
\label{Bloch_Eqs_chi_stoch_res}
\tag{14.15}
\chi(\omega,T)=\frac{\chi^{\rm eq}}{1-i\omega T_1}\,.
\end{equation}\]
The peak occurring in the a.c. susceptibility(14.15) when \(\omega T_1\simeq 1\) has nothing to do with the resonance condition \(\omega=\omega_{\rm L}\) encountered in EPR, NMR or FMR experiments.
To underline this difference the denomination *stochastic* resonance is used to indicate the resonance occurring when \(\omega \, T_1=1\).

Figure14.2 shows the imaginary part of the a.c. susceptibility measured on four crystals of Fe4 with different chemical composition.

## 14.2 Néel-Brown law for nanoparticles

This figure sketches how the scheme of levels of the anisotropy energy associated with a spin \(S=5\) transforms into a continuous function of the polar angle (the angle formed by \(\vec S\) with the easy anisotropy axis) when the quantum mechanical spin operator is replaced by a classical vector. From the perspective of equilibrium thermodynamics, both wells of the energy landscape on the right should be equally probable, irrespectively of the height of the energy barrier separating them (similarly to the pairs of levels \(m_s=\pm 5, \pm4, \dots\) in the quantum mechanical counterpart, on the left). Another way to state that is by saying that the system spends half of the time at the bottom of each well (see Fig.14.3). Particularly relevant is transition rate from one well to the other, i.e., from the state \(\uparrow\) to \(\downarrow\) and vice versa. For \(k_{\rm B}T \ll D\), it can be shown that this frequency has an Arrhenius-like dependence \[\begin{equation} \label{arr_superpara} \tag{14.16} \nu = \nu_0 \exp\left(-\frac{DS^2}{k_{\rm B} T} \right)\,. \end{equation}\] Within the nomenclature of spin relaxation, this frequency equals \(1/T_1\). Since Eq.(14.16) applies to classical spins, it is expected to hold when the thermal energy \(k_BT\) is of the order of – or larger than – the spacing between the two lowest lying levels in zero field but smaller than the total barrier height^{47}. Equation(14.16) also holds for comparatively small nanoparticles containing \(N\) magnetic moments rigidly coupled with each other, provided that \(D\) is replaced by the total anisotropy \(ND\). In the context of magnetic storage, this variation of Eq.(14.16) is used to determine the minimum size that a nanoparticle should have in order that it can keep a record for a predefined (say 10) number of years. This condition on the particle size, via \(N\), defines the so-called

*superparamagnetic limit*.

In the derivation of Eq.(14.16) a fluctuating, random field \(\vec{B}^{\rm rnd}(t)\) is added to the LLG equation that describes the dynamics of a classical spin \(\vec{S}\):

\[\begin{equation}
\label{stoch_LLG}
\tag{14.17}
\dot{\vec S}=- \vec{S} \times \left\{\gamma\left[\vec{B}^{\rm eff} + \vec{B}^{\rm rnd}(t)\right] + \alpha \,\dot{\vec S} \right\} \,.
\end{equation}\]
The effective field appearing in Eq.(14.17) is a generalization of the external field and reads
\[\begin{equation}
\label{effective_field}
\tag{14.18}
\vec{B}^{\rm eff}=\frac{1}{g\mu_B}\frac{\delta \mathcal{H}}{\delta \vec{S}}
\end{equation}\]
where the Hamiltonian \(\mathcal{H}\) may include single-spin terms – like Zeeman and magnetic anisotropy energy – as well as the coupling with other spins of exchange and dipolar origin. In this approach *à la Langevin*, the Gaussian noise \(\vec{B}^{\rm rnd}(t)\) is characterized by the following properties:
\[\begin{equation}
\label{stoch_field}
\tag{14.19}
\langle B^{{\rm rnd},\, \eta}(t)\rangle=0 \qquad\qquad
\langle B^{{\rm rnd},\, \eta}(t)\, B^{{\rm rnd},\, \sigma}(t+\Delta t)\rangle = b \, \delta_{\eta\sigma} \, \delta(\Delta t)
\end{equation}\]
where \(\langle\dots\rangle\) stands for statistical average and \(\eta,\sigma=x,y,z\). To be quantitative, these assumptions hold as long as time correlations in the random field \(\vec{B}^{\rm rnd}(t)\) occur at much shorter time scales^{48}
than the Larmor period \(1/\omega_{\rm L}\). The constant \(b\) appearing in Eq.(14.19) is to be determined by some correspondence principle with the Boltzmann distribution at thermal equilibrium. As for Brownian motion, it turns out to be proportional to temperature.

### Bloch versus stochastic LLG equation

A this point it is worth pinpointing the differences between the descriptions of how a spin system evolves towards thermal equilibrium according to the stochastic LLG equation and the Bloch equation. As illustrated by the following sketch
the starting point for both approaches is the Larmor precession defining the zero-temperature dynamics of a spin system. Treating the spin as a classical vector with constant modulus underlines the stochastic-LLG description, in which the effect of thermal fluctuations is taken into account by averaging over many realizations of noise. Technically this approach requires to solve a stochastic equation. On the contrary, the spin variable is not treated as a classical vector in the Bloch equation. The logical jump in this second equation consists of assuming that a precessional equation of the same form as the one deduced for \(T=0\) holds also for the thermal averages of individual spin components. A relaxation term is then superimposed to this precessional contribution, which finally yields a *deterministic* equation for the out-of-equilibrium magnetization \(\vec M(t)\).

## 14.3 Concluding remarks

Our journey through the principles underlying magnetism in solids started from the right-hand side of Fig. 14.4. About half of the course was devoted to justifying the terms appearing in simple spin Hamiltonians: How magnetic moments associated with unpaired electrons in free ions of transition metals and rare earths are preserved in the solid-state phase; the origin of magnetocrystalline and shape anisotropy; the mechanisms producing exchange interaction, both in the context of intinerant (RKKY) and localized electrons (super exchange, double exchange).With the spin Hamiltonian at hand, we have learned that Boltzmann statistics provides the reference magnetization curve, i.e. \(\vec M(B,T)\), at thermal equilibrium. Some models – defined by the type of spin Hamiltonian and by the dimensionality of the magnetic lattice – are compatible with the occurrence of spontaneous magnetization, which theoretically would be evident as a discontinuity in the *equilibrium* magnetization curve. However, even at a theoretical level, such a discontinuity is predicted to smear out in realistic bulk magnets when dipolar interaction is taken into account. The systems for which this (Griffiths’ theorem) scenario is relevant at thermal equilibrium are those on the left of Fig. 14.4. A uniform magnetization profile minimizes, instead, the free energy of single-domain nanoparticles (Fig. 14.4 middle). Thus, from the point of view of the magnetization profile, these systems would be good candidates for showing a finite spontaneous magnetization and for the observation of critical phenomena. But rigorously, being finite, single-domain nanoparticles should behave as superparamagnets. In reality, dipolar effects and slow relaxation of the magnetization (which may prevent reaching thermal equilibrium) have to be handled with caution in the study of critical phenomena in magnetism. Moving down in scale towards the right in Fig. 14.4 we find molecular clusters, which are too small to represent the thermodynamic limit and, therefore, to sustain spontaneous magnetization: Their equilibrium magnetization curve is certainly not compatible with a discontinuity.

Once the reference states of thermodynamic equilibrium are defined, it is easier to summarize which mechanisms drive the reversal of the magnetization. Starting from the right panel of Fig. 14.4, the Néel-Brown magnetization reversal is expected to be the standard mechanism in molecular clusters, with some accelerations in correspondence to few level anticrossings where resonant quantum tunneling may become sizable. Note that for the observation of genuine quantum phenomena it is crucial that the giant spin of individual molecules be not too large. The Néel-Brown mechanism is also one of the possible ways to reverse the magnetization of single-domain nanoparticles. Other possibilities are non-uniform magnetization reversal or curling. The choice between the possible reversal mechanisms strongly depends on the size and the shape of the considered system. The hysteresis reported on the left panel of Fig. 14.4 was measured on an individual elliptic CoZr sample of size 1\(\mu\)m \(\times\) 0.8\(\mu\)m and 50 nm thick. It exemplifies systems in which nucleating a domain wall is relatively easy because the increase in the exchange energy is compensated by a decrease in the dipolar energy. After domain walls have been created, magnetic domains should expand or shrink in order to adjust to the equilibrium value of \(\vec M(B,T)\). When no external field is applied, by virtue of the Griffiths’ theorem, at thermal equilibrium magnetic domains should realize a configuration in which the total magnetization of the sample vanishes. Whether this configuration is attained in a reasonable time or not establishes if a bulk magnet behaves as a *soft* or as a *hard* magnet. This question involves out-of-equilibrium thermodynamics combined with many extrinsic contributions, such as the pinning of domain walls by defects.

This modest account of the quest which led mankind from understanding the origin of atomic magnetic moments to describing the thermodynamics of bulk magnets is far for being exhaustive. Nevertheless, we hope that it will help you (reader) structure your personal knowledge and foster your curiosity for advanced research in magnetism: there are still plenty of open issues that are waiting for your scientific contribution!

Another way to understand the two different populations is to think of a macroscopic sample containing \(N\) non-interacting spin 1/2 of which \(N_\downarrow(t)\) are in the state \(|\downarrow\rangle\) and \(N_\uparrow(t)\) in the state \(|\uparrow\rangle,\) the corresponding probabilities being \(\rho_{\downarrow\downarrow}(t)=N_\downarrow(t)/N\) and \(\rho_{\uparrow\uparrow}(t)=N_\uparrow(t)/N\).↩︎

The total barrier height is \(DS^2\) for integer spins and \(DS^2-D/4\) for half-integer spins.↩︎

Based on the quantum-mechanical Nyquist formula, the spectrum of thermal fluctuations may be regarded as white up to a frequency of the order of \(k_BT/\hbar\). At room temperature this yields correlation times of the order of 10\(^{-13}\) s, well below a typical precessional period (for instance, about an anisotropy field).↩︎