I've modeled the cooldown of a 2" diameter and 4" diameter Si wafer in Attachments 1 and 2, using the current Megastat model and previous cold head temperature data. The model includes heat leaking into the inner shield enclosure from an aperture, which we currently observe in Megastat cooldowns. (Note how the wafer cools down much faster than the current test mass, due to the very tiny volume.)
The analytic equation for radiative heat transfer in a 2-surface enclosure (formed by the inner shield and Si wafer) is:
This is dependent on properties of inner shield / cold plate, and as such the accuracy of wafer emissivity measurements will be limited by our uncertainty on the inner shield and cold plate emissivities.
As the ratio  approaches 0, the above equation simplifies to:
 . The terms related to the surrounding surface (inner shield) drop out of the equation, and so the smaller the ratio of areas, the less of an impact the inner shield / cold plate emissivities will have on the cooldown. Thus we should seek to minimize the ratio of areas to minimize the uncertainty on eSi.
On the other hand, in this low area ratio limit, the thermal power transfer between the wafer and surrounding inner shield is proportional to the area of the wafer. As the attachments show, the 4" diameter wafer gets colder than the 2". This should be taken into account when determining in what temperature range we would like to fit the wafer emissivity. Larger wafer ---> colder. Do we care about emissivity measurements < 123K? If not, the 2" wafer gets us there. |