The Doppler effect of thermal reactors is due to capture resonances in the non-fissionable fuel isotopes. It considers power excursions without heat removal, which is hypothetical. The Doppler temperature coefficient is:
$$\alpha^D_{T_f}=\frac{\partial \rho}{\partial T_f}$$
The reactivity of the Doppler effect is
$$\rho(t)=\rho_0-\alpha^D_{T_f}(T_f(t)-T_0)$$
With \(T_f\) is the adiabatic approximation of fuel temperature. With the step change in temperature feedback is
$$\frac{d\rho}{dt}=-\alpha^D_{T_f}Hn(t)$$
The equation for heat in the core of the reactor is
$$\frac{dT_f}{dt}=Hn(t)$$
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