# Speed Distribution of The Particles

I want to know the distribution of the particles's speed.

The particles what I mean are nucleons and electrons of element.

Consume there is 1kg of iron on room temperature and it's shape is sphere.

And also it's in the vacuum environment.

Question. How can I get the distribution of particles's speed?

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since you mentioned about electrons in a metal, the distribution of electronic energies is governed by Fermi-Dirac statistics. I have no idea about nucleonic energy distribution. –  Vineet Menon Nov 11 '11 at 8:16
pleasure....... –  Vineet Menon Nov 11 '11 at 8:48
that kind of sh** happens in SE... :(...but its not me!!! –  Vineet Menon Nov 11 '11 at 11:43

The distribution of classical speeds of any collection of particles is always (proportional to)

$$e^{-mv^2\over 2kT}$$

Where k is Boltzmann's constant, T is the absolute temperature, and m is the mass of the particle. This is valid at room temperatures for nucleon speed distribution, since the nuclei are classical (this is the Born Oppenheimer approximation).

For electrons, you are out of luck. These are very quantum. It doesn't even make sense to ask their speeds, because they are in distributed quasiparticle states near a Fermi energy--- they are a cold quantum gas.

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First you have to define what you mean by speed.

A ball of iron is composed by atoms of iron in an amorphous lattice, and it is subject to thermodynamic equations and statistical mechanics equations and quantum mechanical equations. In solids the degrees of freedom of the atoms are vibrational and rotational, so even though they have a "speed" they are not speeding anyplace. In vacuum your iron ball will radiate all the kinetic energy from the vibrational and rotational degrees of freedom and come to a temperature near absolute zero.

Iron also will have electrons freely moving in bands if an electric field is applied, and those electrons will have a velocity distribution dependent on the field applied.

When one comes within the iron nuclei, there are velocity probability distributions given by solutions to the quantum mechanical problem, within each nucleus for the protons and neutrons, and also for the bound electrons. They are not speeding anyplace either.

One can go into individual protons or neutrons and there, there will be quantum mechanical velocity probability distributions for the bound quarks making them up.

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""I think you've translated liberally the speed to velocity."" ROFL! I think You translate Your - maybe- intelligent questions/comments with the aid of babbelfish? –  Georg Nov 11 '11 at 14:27
@kso83o The universe is quantum mechanical in all aspects. Macroscopically we may not see it, but in the microcosm, which is where your question is classified, quantum mechanics rules. Vibrational and rotational degrees of freedom give rise to energy levels. These are the ones that will be radiating by photons away the energy, falling to lower and lower levels until they are at the lowest energy level of that degree of freedom, and the temperature of the solid will be near 0 kelvin. –  anna v Nov 11 '11 at 15:03
@kso83o Then you state in vacuum. In vacuum all the rotational and vibrational kinetic energy of the molecules/atoms that retains the ball at 20C will radiate away because it will be no longer in temperature equilibrium with air. Vacuum has about -273C . –  anna v Nov 12 '11 at 5:09
@kso83o I apologize for the ignorance of other people on this site. They should be ashamed of themselves. –  Marty Green Nov 12 '11 at 15:21