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An implicit assumption in paleointensity experiments is that TRM acquisition is linearly proportional to the applied field. However, both single domain theory Néel (1949) and experimental data (Selkin et al., 2007) provide evidence that some particles can acquire TRM in a non-linear fashion in applied fields that are typical of the geomagnetic field intensity (i.e., $$\lesssim$$ 100 $$\mu$$T).
Single domain theory (Néel, 1949) predicts that TRM is proportional the hyperbolic tangent of the applied field ($$B$$) as described by: \[ TRM=M_{rs}\tanh{\left(\frac{VM_s(T_b)B}{kT_b}\right)}, \] where $$M_{rs}$$ is the saturation remanent magnetization, $$V$$ is the grain volume, $$M_s(T_b)$$ is the saturation magnetization at the blocking temperature ($$T_b$$), and $$k$$ is the Boltzmann constant. For weak fields the linear approximation generally holds true for most SD grains and is a result of the approximation that $$\tanh(x)\approx x$$ for small $$x$$.
In comparison to Néel theory, Selkin et al., (2007) proposed that non-linear TRM acquisition be approximated by: \[ TRM=A_1\tanh{(A_2B)}, \] where $$A_1$$ and $$A_2$$ are scaling coefficients. It can be noted that the linear approximation is valid in the limit as $$A_2$$ tends to zero and that linearity of magnetization with applied field is a special case of the more general non-linear form. This simple approximation assumes that $$A_2$$ is temperature invariant, which in the strictest sense is not true $$\left(A_2 \propto \frac{M_s(T_b)}{T_b}\right)$$. This approximation, however, has been demonstrated to fit real data well (Selkin et al., 2007; Shaar et al., 2010) and the assumption of a temperature invariant $$A_2$$ is equivalent to assuming that the degree of non-linearity is identical for all pTRMs and total TRMs.The non-linear behaviour of a specimen can be determined by imparting the specimen with TRMs acquired in a range of applied field. A best-fit hyperbolic tangent model (of the form given above) can then be fitted to the data to determine the $$A_1$$ and $$A_2$$ coefficients. For an SD specimen in the absence of chemical alteration (i.e., the coefficients $$A_1$$ and $$A_2$$ do not change), the slope of the line ($$\left|b\right|$$) on an Arai plot can be described by: \[ \left|b\right|=\frac{NRM_{Anc}}{TRM_{Lab}}=\frac{\tanh{(A_2B_{Anc})}}{\tanh{(A_2B_{Lab})}}, \] and $$B_{Anc}$$ is: \[\label{eqn:Fa_NLT} B_{Anc}=\frac{ \tanh^{-1}\left({\left|b\right|\tanh{(A_2B_{Lab}) } } \right)}{A_2}. \]
Statistic: $$\delta{TRM_{NLT}}$$
Report to 1 d.p.
To test for alteration during the measurement TRM acquisition a repeat remagnetization step is performed in the first laboratory field, which is typically the same field as used for the paleointensity experiment (i.e., the final TRM acquisition in $$B_{Lab}$$). $$\delta{TRM_{NLT}}$$ is the difference between the TRM acquire in first heating in $$B_{Lab}$$ ($$TRM_{B_{Lab}}$$) and the second heating in $$B_{Lab}$$ ($$TRM'_{B_{Lab}}$$) normalized by $$TRM_{B_{Lab}}$$. \[ \delta{TRM_{NLT}}=\frac{\left|TRM_{B_{Lab}} - TRM'_{B_{Lab}}\right|}{TRM_{B_{Lab}}}\times{100} \]
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