3.2.2 Low Temperature Behaviour

Typically, the low temperature behaviour of (3.9) is approximated by an Arrhenius law,

while this certainly accounts for the majority of the low T dependence, it is not the complete dependence. Perfect Arrhenius behaviour is only rigorously found if the integrand of (3.9) is gapped but otherwise featureless in energy. The strongest energy dependence one might expect in the integrand (for a gapped density of states) is that of BCS (3.8). Allowing for finite $\alpha_m$, the singular behaviour of the square of the BCS density of states is avoided, and the integral can be expanded at low T in terms of modified Bessel functions, giving a temperature dependent prefactor to the Arrhenius dependence of T^-1/2. This is analogous to the temperature dependence of the penetration depth[103,106,107]. T^-1/2 is a weak function of temperature compared to (3.10), but it does lead to a significant bias in the energy gap extracted using (3.10). For example, if one fits (3.10) to data that varies as $T^{-1/2}\exp{(-1.76T_c/T)}$, over a range of reduced temperature t = T/T_c of 0.25-0.5 (typical for many NMR studies), one finds $\Delta_0 = 1.56kT_c$,and the Arrhenius plot doesn't deviate noticeably from linear. While the specific T^-1/2 dependence is highly idealized, this example illustrates the dangers of using a simple model such as (3.10) especially over a restricted range in temperature. The low temperature behaviour in cases less ideal than (3.8) will be determined by balancing the contributions of both the peak in g(E) at $\Delta$ (if it is present), and any finite g(E) within the ``gap'', with the latter always dominating at the lowest temperature because of the exponential weighting of the Fermi factor. We will consider more realistic models for g(E) in the next section.