Intermolecular mechanism for multiple maxima in molecular dynamic susceptibility

A simple two-level system is introduced to demonstrate the existence of the intermolecular mechanism of the appearance/disappearance of multiple maxima in molecular dynamic susceptibility at low temperature. With a minimum of two relaxation processes, quantum tunneling induced by the nuclear spin bath, and the direct process due to the spin-phonon interaction, we prove that the existence of a sufficiently wide dipolar field distribution in the experimental sample is the main cause of this phenomenon in zero or weak applied dc field. The correlations between the phenomenon and the applied dc field, temperature, or magnetic dilution of the sample are also investigated. Application to several experimental systems has shown a good agreement between the proposed theory and experiments. Cases with a more complex multiplet spectrum, more relaxation processes, and the possibility of more maxima in the molecular dynamic susceptibility are discussed.

Figure 1

Out-of-phase susceptibility of a non-Kramers polycrystalline sample (${\chi }_0$ units) without an internal dipolar field but with an applied dc field $h_{dc}=1$ (a), 3 (b), 6 (c), and 10 (d).

Figure 2

Out-of-phase susceptibility of a Kramers polycrystalline sample sample (${\chi }_0$ units) without an internal dipolar field but with an applied dc field $h_{dc}=1$ (a), 3 (b), 6 (c), and 10 (d).

Figure 3

Out-of-phase susceptibility of a non-Kramers single-crystal sample in $x^2{\chi }_0$ units with $h_{dc}=0$ and $h_{dm}=0.1$ (a), 1 (b), 3 (c), and 10 (d).

Figure 4

Out-of-phase susceptibility of a non-Kramers single-crystal sample in $x^2{\chi }_0$ units with $h_{dc}cos\theta =3$ and $h_{dm}=0.1$ (a), 1 (b), 3 (c), and 10 (d).

Figure 5

Out-of-phase susceptibility of a Kramers single-crystal sample in $x^2{\chi }_0$ units without an external field and $h_{dm}=0.1$ (a), 1 (b), 3 (c), and 10 (d).

Figure 6

Out-of-phase susceptibility of a Kramers single-crystal sample in ${\chi }_0$ units with $h_{dc}=3$ and $h_{dm}=0.1$ (a), 1 (b), 3 (c), and 10 (d).

Figure 7

(a) Out-of-phase susceptibility of ${\left({\mathrm{Ph}}_4\mathrm{P}\right)}_2\left[\mathrm{Co}{\left(\mathrm{SPh}\right)}_4\right]$ at $T=2$ K under applied dc field from 0 to 1000 Oe in 100 Oe increments (reprinted with permission from Zadrozny and Long [34], copyright 2011 American Chemical Society). (b) The same quantity generated from our model using an averaging of Eq. (11) for Kramers powder sample, an addition of the second-order Raman process rate ${\mathrm{\Gamma }}_{\mathrm{Raman}}=C_{\mathrm{Raman}}{T}^9$ into the expression of ${\mathrm{\Gamma }}_{m{m}^{\prime }}$, Eq. (3), and parameters $\alpha =0$, $C_{\mathrm{Raman}}=0.00185{\mathrm{s}}^{-1}{\mathrm{K}}^{-9}$, $H_n=75\mathrm{Oe}$, $H_{dm}=150\mathrm{Oe}$, ${\mathrm{\Gamma }}_{\mathrm{tn},0}=63{\mathrm{s}}^{-1}$, and ${\chi }_0=2.9{\mathrm{cm}}^3{\mathrm{mol}}^{-1}$.

Figure 8

(a) Frequency dependence of the out-of-phase susceptibility of $\left[\mathrm{Co}{\left({\mathrm{Tp}}^{*}\right)}_2\right]$ under 400 Oe applied dc fields for various temperatures from Li et al. [16], reproduced by permission of the Royal Society of Chemistry. (b) The same quantity generated from our model using an averaging of Eq. (11) for Kramers powder sample, an addition of the second-order Raman process rate ${\mathrm{\Gamma }}_{\mathrm{Raman}}=C_{\mathrm{Raman}}{T}^7$ into the expression of ${\mathrm{\Gamma }}_{m{m}^{\prime }}$, Eq. (3), and parameters $\alpha =0$, $C_{\mathrm{Raman}}=0.025{\mathrm{s}}^{-1}{\mathrm{K}}^{-7}$, $H_n=125\mathrm{Oe}$, $H_{dm}=600\mathrm{Oe}$, ${\mathrm{\Gamma }}_{\mathrm{tn},0}=100{\mathrm{s}}^{-1}$, and ${\chi }_0=5.7{\mathrm{cm}}^3{\mathrm{mol}}^{-1}$.

Figure 9

(a) Frequency dependence of the out-of-phase susceptibility of $\left[\mathrm{Co}{\left({\mathrm{Tp}}^{*}\right)}_2\right]$ with 10 times magnetic site dilution under 400 Oe applied dc fields for various temperatures from Li et al. [16], reproduced by permission of the Royal Society of Chemistry. (b) The same quantity generated from our model using an averaging of Eq. (11) for Kramers powder sample, an addition of the second-order Raman process rate ${\mathrm{\Gamma }}_{\mathrm{Raman}}=C_{\mathrm{Raman}}{T}^7$ into the expression of ${\mathrm{\Gamma }}_{m{m}^{\prime }}$, Eq. (3), and parameters $\alpha =0$, $C_{\mathrm{Raman}}=0.025{\mathrm{s}}^{-1}{\mathrm{K}}^{-7}$, $H_n=125\mathrm{Oe}$, $H_{dm}=60\mathrm{Oe}$, ${\mathrm{\Gamma }}_{\mathrm{tn},0}=100{\mathrm{s}}^{-1}$, and ${\chi }_0=5.7{\mathrm{cm}}^3{\mathrm{mol}}^{-1}$.