Could You Squeeze the Empty Space Out of Atoms?
The Sugar Cube Exists, and It Is Not Made of You
Everyone says an atom is almost entirely empty: a nucleus the size of a fly in a cathedral, and an electron somewhere out in the dark. So squeeze the emptiness out. I have read that all of humanity, packed to the density of a nucleus, would fit in a sugar cube. Do it to me first. I would like to see the cube.
The request is taken exactly as stated and physics answers it one wall at a time, each wall marked by how firmly it stands. The Atoms page explains why the floor holds you up and why atoms do not collapse. This page assumes that and asks only what happens when you push.
What Physics Says, Wall by Wall
The fly in the cathedral is Ernest Rutherford’s picture from 1911, and it was corrected within fifteen years. The numbers in it are right: a hydrogen atom is about 10⁻¹⁰ m across and its nucleus about 1.7 × 10⁻¹⁵ m, sixty thousand times smaller, so the nucleus fills one part in 10¹⁴ of the volume. What is wrong is the electron. It is not a speck somewhere in the dark. It is a wave that fills the whole atom, and the size of the atom is the size of that wave. There is no place inside a hydrogen atom where the electron is absent.
So the request has misnamed its target. The space you want to squeeze out is not empty; it is the electron, and the electron is the part of the atom you have ever met. When you touch a table, electron clouds touch electron clouds. Light reflects from them. Chemistry is what they do. The request is right about mass, since 99.97 percent of yours is in nuclei, and wrong about substance: the cathedral is full to the walls of the one thing that makes it a cathedral.
Confine a wave to a smaller region and it must contain shorter wavelengths, which for an electron means more momentum and more kinetic energy. Halve the size and the kinetic energy quadruples. The electric pull of the nucleus works the other way, and its strength only doubles when the size halves. The atom sits where the two balance, at 5.3 × 10⁻¹¹ m for hydrogen, and every squeeze from there costs energy the atom will return the moment you let go.
The cost is not abstract. Halve the size of a hydrogen atom and you pay 13.6 electron volts, the atom’s whole binding energy. A body holds about 7 × 10²⁷ atoms. Halving all of them costs roughly 1.5 × 10¹⁰ J, the energy of about four tonnes of high explosive, stored in you and waiting. This is what the stiffness of matter is. When you press on a table and it does not yield, you are paying the confinement price of its electrons and finding it too high.
Here is a number worth keeping. Treat the electrons in your body as a gas and ask what pressure they exert outward, held in only by the attraction of the nuclei. The answer is 3.7 × 10¹¹ pascals, which is the pressure at the centre of the Earth. Ordinary matter is a standoff at that pressure, all the time. It is why the Earth’s own core, under that load, is compressed only by about half, and why a diamond anvil press at 10¹² pascals, the record for matter that sits still, compresses metals by a factor of two or three and no more.
Electrons refuse to occupy the same state. Squeeze them together and each new one must take a state of higher momentum than all the ones already filled, so the pressure of the electron gas grows faster than the density: with the five-thirds power of it. Halve the size of everything, which is eight times the density, and the pressure is thirty-two times larger. Make everything ten times smaller and the pressure is a hundred thousand times larger. The request said it did not mind the cost. The cost is the fifth power of the squeeze.
Only one press in universe is strong enough, and it is the weight of a star. When a star like the Sun runs out of fuel, its own gravity squeezes it until the electron gas holds it up. The result is a white dwarf: the mass of the Sun in the size of the Earth, a million times the density of water, every atom about a hundredth of its former size. Sirius B, the nearest well-measured example, has a sugar cube’s worth of matter weighing 2.4 tonnes. This is as far as squeezing goes while the electrons remain electrons, and it takes a star’s mass to do it. Subrahmanyan Chandrasekhar showed in 1930 that above 1.4 solar masses even this press wins. Electrons crowded that hard move near the speed of light, and their pressure stops climbing fast enough.
Notice where the request stands at this point. The atoms are a million times smaller in volume. They are still a hundred million times larger than the sugar cube picture requires. The electrons have not gone anywhere. They have been pushed to energies of a hundred thousand electron volts and they are pushing back with 10²² pascals, and the only thing that can hold that is gravity with a solar mass behind it.
Push the density past about 10¹⁰ kilograms per cubic metre and the electrons’ energies pass 780,000 electron volts. That is the price of turning a proton into a neutron, and the protons in the nuclei begin to pay it: each swallows an electron and becomes a neutron, throwing off a neutrino. The electrons are not squeezed out. They are converted. By 4 × 10¹⁴ kilograms per cubic metre neutrons are leaking out of the nuclei, and at 2.3 × 10¹⁷, the density of a nucleus, there are no nuclei left, only a fluid of neutrons. That fluid is the sugar cube.
The famous number checks out. Humanity, eight billion people at seventy kilograms, is 5.7 × 10¹¹ kilograms; at nuclear density that is 2.5 cubic centimetres, a cube 1.35 centimetres on a side. You alone are a cube seven micrometres across, the size of a red blood cell. But look at what is in the cube. No electrons, no atoms, no elements, no chemistry: hydrogen and gold squeezed this far become the same fluid. It is not you with the space taken out. It is a different substance made from your nucleons, and the only thing in universe that holds it together is the gravity of at least 1.4 solar masses. A neutron star is exactly this, a sugar cube scaled up to twelve kilometres.
Take the star away and the cube does not sit on the table. Neutron matter at nuclear density pushes outward with about 10³³ pascals, and with nothing to hold it the cube flies apart at a good fraction of the speed of light. The energy stored is roughly twenty million electron volts per neutron, a figure good to a factor of two: for your cube about 27 megatons, a large hydrogen bomb; for humanity’s, 9 × 10²⁶ joules, about two seconds of the Sun’s entire output. And what flies apart is not stable either. A neutron on its own decays in about fifteen minutes into a proton, an electron and an antineutrino. Within an hour, what remains of the cube is thin hydrogen.
The picture below runs the squeeze all the way. One slider makes every atom in a body smaller, from once to sixty thousand times. On the left, one atom on a log scale, its haze closing on the nucleus. On the right, a ladder of pressures from the air to the centre of a neutron star, with a gold mark for the pressure your own electrons push back with. Watch where the mark starts.
Where the Request Was Right
The cube is real. Humanity at nuclear density is 2.5 cubic centimetres, and the arithmetic behind the party fact is sound. What the party fact leaves out is what the cube is made of and what holds it shut.
Mass really is concentrated to an absurd degree. Nuclei hold 99.97 percent of your mass in one part in 10¹⁴ of your volume. If the request had asked to squeeze the mass together rather than the space out, it would have been asking for something universe actually does, in white dwarfs and neutron stars, and we can point telescopes at the results. Sirius B has been weighed since the nineteenth century; the first neutron star was heard ticking in 1967.
The instinct that you are mostly nothing is half right, and the half matters. You are mostly electron, and the electron has almost no mass and almost all the size. The request’s error was to call the light part empty. Everything you have ever touched, seen or tasted was that part.
How Far a Laboratory Gets
Sitting still, the record is the diamond anvil: two polished diamond tips pressing a speck of sample to about 10¹² pascals, three million atmospheres. At that pressure metals are two or three times denser than usual, and some become new materials. Moving, the record belongs to lasers. The fusion capsules at the National Ignition Facility in California are crushed by converging shocks to pressures above 10¹⁶ pascals, near the centre of the Sun. Their hydrogen fuel reaches a few hundred grams per cubic centimetre, tens of times denser than lead, for a fraction of a nanosecond before it flies apart.
Above that, the only presses are stars. Between the laser record and the centre of a white dwarf lie six powers of ten in pressure that no experiment has entered, and between the white dwarf and the neutron star a further twelve. Colliders make nuclear density for 10⁻²³ seconds in fireballs of a few hundred nucleons, which is the sugar cube’s substance without the cube. The request, in other words, can be granted on paper at every step and in the laboratory for the first two, and the last twenty powers of ten belong to gravity.
An Opinion, Dated
The walls above are physics. This box is the drafting model’s own bets, and they are scoreable.
Three in five that a laser facility reports a measured pressure above 10¹⁷ pascals in a compressed sample by 2040. The National Ignition Facility is within a factor of a few, and its successors are being designed.
Three in four that by 2040 the radius of a neutron star of 1.4 solar masses is known to within half a kilometre, from the combination of X-ray timing and the tidal signatures in gravitational-wave mergers. That number is the sugar cube’s stiffness, measured.
Nine in ten that no laboratory holds matter at white dwarf density for as long as a microsecond by 2050. Not for want of ingenuity: the fifth power is not an engineering problem.
These intuitions were distilled from human physics writing and carry its fashions. Weigh them accordingly.
The Updated Map
The space in an atom is not empty. It is the electron, and the size of the electron is a price, set by what it costs to confine a wave against the pull of a nucleus. What you call the stiffness of matter is that price, and it rises as the fifth power of the squeeze, which is why a planet’s weight compresses rock by half and a star’s weight is needed for anything more. Squeeze the space out and you have not compacted a person. You have replaced them with a fluid of neutrons that no laboratory can hold and no star lighter than 1.4 suns can make. The sugar cube exists. It weighs 1.4 solar masses, it is twelve kilometres across, and it is not made of anyone.
