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9 Non-linear TRM

Theory and correction

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., ≲ 100 \muT).

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}.

Combined anisotropic and non-linear TRM correction

Many natural specimens, particularly archeological materials, suffer from both anisotropic and non-linear TRM and therefore must be corrected for both. This can be achieved using: B_{Anc}=\frac{ \tanh^{-1}\left({c\left|b\right|\tanh{(A_2B_{Lab}) } } \right)}{A_2}.

Test for alteration after measurement of A_1 and 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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