Fine calcium carbonate powder from eggshells

My goal is to create a very fine calcium carbonate powder using only common materials and ingredients, no specialized chemicals, equipment or tools. I start by placing an egg in vinegar, and letting that sit until the eggshell is gone. Assuming there's not too much acetic acid, this should give me calcium acetate and water, right?


Next, I bake some baking soda at 200F for an hour or two, hopefully this produces sodium carbonate, which is more water-soluble than sodium bicarbonate. I dissolve this in water until it is saturated. (my goal here is to ensure that any powder produced is the target material, without precise measurements, so we combine two clear liquids, rather than adding a powder into a clear liquid)


Finally, I add the dissolved sodium carbonate to the calcium acetate solution until it looks like nothing is happening anymore, which should precipitate ~pure calcium carbonate, right?


It seems that calcium carbonate takes different forms, and I'd like to produce the variety that has certain spectral qualities (calcite?). Namely, reflects the majority of the solar spectrum (.3-2.5 μm) and emits in the atmospheric window spectrum (8–13 μm). Will this process work?


