Ac susceptibility

The magnetisation MAC cannot follow the incident field immediately and is shifted by a phase angle 0, so that 

Thus the magnetic susceptibility in an alternating field, the AC susceptibility, can be viewed as a complex quantity 

The components of the AC susceptibility depend upon the frequency of the applied field. When the frequency of the field is low, the magnetisation can readily follow the applied field so that the low-frequency limit corresponds to the isothermal susceptibility 

This is comparable with the susceptibility measured in static fields. The greater the frequency, the lower the AC susceptibility, which in the opposite limit approaches the adiabatic value 

The recovery of a perturbed system to a new equilibrium is described by introducing the relaxation time x 

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Similarly, a linear Dy3 compound (9, Figure 3.7b) with a Dy-Dy-Dy angle of 165.34° was reported by Fang and coworkers [41]. Here the Dy3 exhibits obvious SMM behaviour as evidenced by well-resolved out-of-phase ac susceptibility maxima between 15 and 351<. 

In addition, three other Dy4 complexes (17-19 Figure 3.13) showing a planar motif were reported in 2010 and 2011[46—48]. Under zero dc field, compounds 17 (Figure 3.13a) and 18 (Figure 3.13b) show weak SMM behaviour with small effective barriers of 6.2 and 4.51<, respectively. In contrast, the ac susceptibility signals without maxima were observed under zero dc field in compound 19 (Figure 3.13c). However, the application of a dc field leads to the obvious enhancement of its SMM behaviour, and an effective barrier of 221< (r0 = 3.66 x 10 6s) can be obtained. 

In 2012, Liao and coworkers [52] reported a Dy6 octahedron (24, Figure 3.17), seen as two edge-sharing [(p4-0)Dy4] tetrahedra, with two tail-to-tail calixarene ligands. The ac susceptibilities show obvious temperature-dependent both in-phase (x ) and out-of-phase (x") signals under a zero dc field, indicating multiple relaxation SMM behaviour. 

The structures and dynamic magnetic behaviours of 31-Dy4 and 31-Tb4 have been presented in Figure 3.22. Four lanthanide ions are aggregated in the grid-like metal core by a central p4-S and eight peripheral i2-S atoms from ethanethiol ligands [38]. The individual lanthanide centres occupy distorted six-coordinate [LnNS5] octahedral coordination environments. Ac susceptibilities measurements reveal pronounced temperature dependence with a series of maxima below 28 K, typical for SMM behaviour, in complex 31-Dy4. Furthermore, an... 

Figure 5.3 Arrhenius plot extracted from ac susceptibility data of Er(trensal) (indicated as 1. V designates the isostructural, but diamagnetic analogue Y(trensal)). (Reprinted with permission from [13]. Copyright 2014, Royal Chemical Society.)...

Figure 10.1 Temperature dependence of imaginary parts of the ac susceptibility (real parts in insets) recorded on powder samples of 1 at different frequencies as indicated, at...

Figure 10.2 Arrhenius plot of the natural logarithm of the relaxation time extracted from the ac susceptibility data as a function of the inverse temperature for 1 at different external fields as indicated. (Reprinted from Ref. [6]. Copyright (2009) American Chemical Society.)...

Table 10.2 Overview of energy barriers and pre-exponential factors derived from Arrhenius fits of the ac susceptibility data for single-molecule magnets 1-15. 

Figure 10.7 In-phase (top) and out-of phase (bottom) components of the ac susceptibility measured in an applied dc field of 2kOe and an ac field of 3.5 Oe for powder...

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