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I am trying to find the best insulation for my apartment. The limitation is that space between the drywall and the ceiling is 1/2". I live on the top floor of my building.

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Last time I checked, polyisocyanurate has the best insulation value per inch (R-6.5) of all the conventional building insulation products, and it is available in 1/2" sheets.

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Likely the best insulator that you could use would be a vacuum. You can purchase "vacuum insulating panels", but they will be difficult to install, expensive, and might end up being too thick. The panel would get you to at least R-20, and probably higher. The material that makes the most sense for you to use is polyisocyanurate. At 1/2 in thickness, it will provide approximately R-3.2. Be sure to seal between the boards, as air leaks can cause quite substantial heat loss.

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Silica Aerogel products should beat polyisocyanurate; 10.3 per inch in commercially-available products.

Polyisocyanurate's performance decreases over time; The manufacturers and the NCRA factor this into the R-Value of Polyisocyanurate products differently.

Vacuum Insulating Panels can use Aerogels as a filler, so you still get about R9 even if you destroy the vacuum with a nail or screw.

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