What is the densest substance in the world?

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About the best you can do with a substance that is stable at STP and thus to be found on Earth is with an alloy of osmium and iridium, which if proportioned correctly and crystallized correctly, has a density of around 22.7 grams per cubic centimeter. You can theoretically optimize it just a little more still, possibly passing 23 g/cm^3 by purifying the heaviest naturally occurring isotopes of each one, osmium-192 and iridium-193, though that would be an awful lot of trouble. It is not known exactly what mix of elements and crystal structure attains the absolute highest possible density. For instance some elaborate repeating structure involving all known elements that have densities over 20 grams/cm^3, namely osmium, iridium, platinum, rhenium and even neptunium, if you're willing to let it be radioactive, could do better than a binary mix of osmium and iridium. It is also possible to attain much greater densities if a solid is subjected to pressures exceeding its bulk modulus, as is for example the plutonium core of an implosion type A-bomb which is momentarily compressed by powerful explosions on all sides to densities in the range of 40 grams/cm^3, twice plutonium's regular density. As to whether you can make a shell hard enough (perhaps carbon nanotube composite) to compress some combination of materials that already have a high density and relatively low bulk modulus, and thus have a package of relatively small size that is significantly higher density than osmium or iridium, the answer to that lies in how large you're willing to have that package be. As fun as it would be to hand the postman a flat rate envelope that weighs 100 million tons, however, the jury says "it's not likely".

If we are off of the assumption that it has to be stable at STP or even stable in a vacuum, then of course the material comprising a neutron or quark star is the densest substance. It is widely believed that these objects have an "atmosphere" which is literally skin deep, followed by a crust of ordinary matter (and by ordinary, I mean it can't be that ordinary since the typical temperature at the surface of a typical neutron star surpasses 100000 K and the magnetic field density is so strong, atoms are crushed from spheres to long strings by the field lines) and then degenerate matter where electrons are partially crushed into the nuclei and mixed in with free neutrons, and then maybe a kilometer down, you get to neutronium, which some say may be pure neutrons, others that there may be free quarks mixed in (making it a 'quark star'). Which amazingly enough is actually a liquid, despite how you may have been misled by Star Trek to think it is an incredibly strong solid. But it takes a lot of pressure to maintain this state. It is a liquid such that if you took a teaspoon of it and transplanted it to Earth, not only would it have a mass of 1.13 billion tons, a little more than the weight of a cubic kilometer of seawater, but it would explode with such energy as it decomposed back to an appreciably smaller mass of regular matter that the oceans would boil and all life in the top mile of the Earth's crust would be turned to ash. No wonder Mary Poppins preferred a spoonful of sugar. How much pressure does it need to maintain this state? Consider that the top kilometer of the star before you get to the neutronium is still degenerate matter which already averages to millions of grams per cubic centimeter, hundreds of thousands of times denser than osmium, but all that mass is additionally under gravity which runs around 100 billion times that at the surface of the Earth. The pressure needed to bring matter to that state in terms of a sound wave would be something on the order of 600 or 700 decibels. So the next time someone invites you to a rock concert in the core of a neutron star, just say no. Or something clever like "I can't take that kind of pressure!" Incidentally, if you did have an inch-sized sphere of neutronium that was somehow stable and didn't explode with cataclysmic power, not only would its gravity be appreciable, but you would be pulled towards it with one Earth-gravity if you were 20 feet away or so. And if you got any closer, things would very quickly get messy.
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