A second question regarding target temperatures in PSII implants is the existence of gradients in the targets or base. The size of gradients in quenched materials (cooled only by convection) is determined by a dimensionless number known as the Biot number[6]. This is given by
where is the heat transfer coefficient governing the convective
cooling and
is the characteristic dimension of the object being
cooled. For thin samples
is the sample thickness, while for
cylinders
is the cylinder radius. For most PSII applications, the
base is thin and it radiates from both
sides, so
is taken to be the thickness of the base.
If the Biot number is much less than 1, then the cooling is limited by the convection and the gradients in the object will be small. We can use a similar process to study PSII target temperature gradients by converting the radiation and cooling rates into an equivalent heat transfer coefficient. In this case, the Biot number will be strongly temperature-dependent because of the radiative cooling. In our case, the effective heat transfer coefficient is given by:
where is the effective heat transfer coefficient,
is the
equivalent heat transfer coefficient for conduction from the base
plate to the chamber, and
is the equivalent heat transfer
coefficient for the radiative cooling. For most PSII applications the
cooling is dominated by radiation, so the effective Biot number can be
approximated as
To determine the equivalent heat transfer coefficient, we start with the radiation relation
and then factor it to give
Comparing this to the usual heat transfer coefficient relation
gives
The effective Biot number thus becomes
where is the base and/or target thickness, and the form factor
is approximately 1. Therefore,
This effective Biot number is plotted in Figure 4 as a function of temperature for different materials. The emissivity is taken to be 1.0, the target thickness is taken to be 1 cm, and the sink temperature is assumed to be 300 K. (The Biot number will be even smaller for realistic emissivities and for heated chamber walls). It is seen that the Biot number is much less than 1 for all the cases considered, so gradients can be ignored when considering the thermal effects of PSII on typical targets with typical heat loads. It should be noted, though, that an actively cooled target or base may exhibit noticeable gradients.
