(i) Assume that the interaction between turbulent modes and both thermal and CRs is fully collisionless. This happens when particles' collision frequency in the ICM is {{\omega}_{\text{ii}}}<\omega =k{{c}_{s}}

for example this is the case where ion-ion collisions in the thermal ICM are due to Coulomb collisions. Under this condition the damping of compressive modes is dominated by TTD with thermal particles (equation (, 4: #ppcfaa0304eqn004)) [, 16: #ppcfaa0304bib016]. For large {{\beta}_{\text{pl}}} the dominant damping rate is due to thermal electrons and can be approximated by {{\Gamma}_{\text{e}}}\sim {{c}_{s}}k\sqrt{3\pi ({{m}_{\text{e}}}/{{m}_{\text{p}}})/20{{\mu}^{2}}}\exp (-5({{m}_{\text{e}}}/{{m}_{\text{p}}})/3{{\mu}^{2}})(1-{{\mu}^{2}}). Combining pitch-angle averaging of this expression with equation (, 7: #ppcfaa0304eqn007) gives {{k}_{\text{cut},\text{K}}}\sim {{10}^{4}}{{k}_{0}}M_{0}^{4}.Assuming a Kraichnan spectrum of magnetic field fluctuations, {{W}_{\text{B}}}\propto {{k}^{-3/2}}, the resulting acceleration time-scale (from equation (, 2: #ppcfaa0304eqn002)) is

\begin{eqnarray}&&{{\tau}_{\text{acc}}}=\frac{{{p}^{2}}}{4{{D}_{pp}}}\simeq 2.5{{\left\langle \frac{{{\beta}_{\text{pl}}}\left|{{B}_{k}}\right|^{2}}{16\pi W}\right\rangle}{}^{-1}}f_{x}^{-1}{{\left(\frac{{{M}_{0}}}{1/2}\right)}^{-4}}\left(\frac{{{L}_{0}}/300~\text{kpc}}{{{c}_{s}}/1500~\text{km} ~{{\text{s}}^{-1}}}\right)~(\text{Myr})\end{eqnarray} \tag{ 8 } where \langle \ldots \rangle \sim 1 independent of scale, x = cs/c and

\begin{eqnarray}&&{{f}_{x}}=x\left(\frac{{{x}^{4}}}{4}+{{x}^{2}}-\left(1+2{{x}^{2}}\right)\ln (x)-\frac{5}{4}\right)\simeq 0.02.\end{eqnarray} \tag{ 9 } For typical conditions this scenario predicts acceleration time-scales of about 100 Myr that are sufficient to explain radio halos, and is commonly adopted to calculate turbulent acceleration in the ICM [, 7: #ppcfaa0304bib007,, 16: #ppcfaa0304bib016,, 20: #ppcfaa0304bib020,, 47: #ppcfaa0304bib047]

(ii) The other possibility is that thermal particles do not contribute very much to the collisionless dampings. The ICM is a weakly collisional high-beta plasma that is unstable to several instabilities (see section, 2: #ppcfaa0304s2). Scattering induced by magnetic field perturbations driven by instabilities may increase the collision frequencies in the thermal plasma, because charged particles can be randomised if they interact with the perturbed magnetic field. This process can be viewed as the collective interaction of an individual ion with the rest of the plasma. Under this condition one may think that the interaction between modes and thermal particles behaves as collisional, 2: #ppcfaa0304fn02. In this case the dominant collisionless damping in the ICM is due to the CRs [, 30: #ppcfaa0304bib030], and combining equations (, 5: #ppcfaa0304eqn005) and (, 7: #ppcfaa0304eqn007) we have

\begin{eqnarray}&&{{k}_{\text{cut},\text{K}}}\simeq {{\left(\frac{18}{{{\pi}^{2}}}\right)}^{2}}M_{0}^{4}f_{x}^{-2}\frac{{{\left(\rho c_{s}^{2}\right)}^{2}}}{{{\left[c\mathop{\int}^{}{{p}^{4}}\text{d}p\frac{\partial f}{\partial p}\right]}^{2}}}{{k}_{0}}.\end{eqnarray} \tag{ 10 } Under typical conditions in the ICM this is {{k}_{\text{cut},\text{K}}}\sim 1000\,{{k}_{0}}M_{0}^{4}{{\left({{\epsilon}_{\text{ICM}}}/{{\epsilon}_{\text{CR}}}\right)}^{2}}, where the ratio of thermal and CR energy densities is {{\epsilon}_{\text{ICM}}}/{{\epsilon}_{\text{CR}}}\sim 100. It implies that the turbulent cascade reaches scales that are much smaller than those in case (i) and consequently the acceleration rate is higher (equations (, 2: #ppcfaa0304eqn002) and (, 10: #ppcfaa0304eqn010)):

\begin{eqnarray}&&\begin{array}{*{20}{l}}{{\tau}_{\text{acc}}}=\frac{{{p}^{2}}}{4{{D}_{pp}}}\simeq 6{{\left\langle \frac{{{\beta}_{\text{pl}}}\left|{{B}_{k}}\right|^{2}}{16\pi W}\right\rangle}{}^{-1}}{{\left(\frac{{{M}_{0}}}{1/2}\right)}^{-4}}\left(\frac{{{L}_{0}}/300\,\text{kpc}}{{{c}_{s}}/1500~\text{km} ~{{\text{s}}^{-1}}}\right)\\ \qquad \ \,\times \left(\left[\frac{c\mathop{\int}^{}{{p}^{4}}\text{d}p\frac{\partial f}{\partial p}}{\rho c_{s}^{2}}\right]/0.04\right)(\text{Myr}).\end{array}\end{eqnarray} \tag{ 11 } The acceleration efficiency is inversely proportional to the energy density of CRs. In this scenario CRs drain energy efficiently from the turbulent cascade

however as the CR energy density increases, the acceleration efficiency is reduced. From the practical point of view this is simply because CRs can obtain a constant energy flux from turbulence, implying that the effect on their spectrum is smaller for increasing values of the CR energy density. As a consequence of this back reaction, fast, acceleration cannot be maintained for long time. Assuming typical conditions in the ICM, the value of the acceleration time averaged on a time-period of few hundred megayears is generally found to be {{\tau}_{\text{acc}}}\sim 10 Myr [, 30: #ppcfaa0304bib030]. This is up to ten times more efficient than in case (i), resulting in harder spectra of both CRe and synchrotron radiation [, 20: #ppcfaa0304bib020,, 30: #ppcfaa0304bib030]