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I'm working in gravitational physics, unfortunately my solid state physics lecture is already a while a go. My question will probably sound quite trivial to solid state physicists, but I couldn't find a fully satisfactory answer, neither in textbooks nor on Google (which is probably just because I don't really no what to search for).

If I have a crystal (say Si, for example) at low temperature (1K or lower). What is the spatial probability distribution for the single atoms in the crystal?

My first guess was something like Einstein's model with independent harmonic oscillators, but if the temperature is low, this doesn't seem to make sense. Even when all oscillators are in the ground state, their energy will be much larger than kT.

So, if one can assume that the wave-function separates in the wave-functions of the single atoms, I guess, one would have to know the actual N-particle wave-function and calculate the marginal distribution for one atom?

I'm sure this is just a textbook problem and I can find the answer somewhere? I would be very thankful if someone could help me, even if it's just giving me the title of a book where I can find this.

Edit:

To be a bit more precise, what I need is this:

Say we have a crystal consisting of N atoms at coordinates $\vec x_1$, ... $\vec x_N$. The crystal is in thermodynamic equilibrium at some small temperature T. The crystal will then be in some state with the spatial wave-function $\psi(\vec x_1,\vec x_2,...,\vec x_N)$. Once this wave-function is given, I could simply calculate the density

$\rho(x_1) = \int d^3 x_2 \,d^3 x_2 \, ... d^3 x_N \, \vert\psi(\vec x_1,\vec x_2,...,\vec x_N)\vert^2$

However, getting the wave-function is probably not so easy? So the question then is, if there is a good approximation/effective model to obtain $\rho(x_1)$, $\rho(x_2)$, ... that works well for small temperatures?

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You are right that the Einstein model will break down if you are looking at low temperature. The next level of approximation, which does better, would be the Debye model. If you look in any intro. statistical physics book (I recommend Schroeder, Introduction to Thermal Physics if you are undergrad, or Kittel's Intro. to Solid State Physics) you will find a good introduction to the topic.

This won't (easily) give you the distribution of particle positions. It will give you populations of phonons. Really, the distribution of position of a single atom isn't very meaningful. Position relative to what? The thing you are probably looking for is the "pair correlation function" which holds information about the distribution of how particles are arranged relative to their nearest neighbours. You won't find this in an intro. treatment, but you'll need to at least know your way around the Debye model before you'll be able to understand how we think about pair correlation functions.

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  • $\begingroup$ "Position relative to what?" The centre of mass of the crystal. Thank you for your answer. I know the Debye model, in principle, but just didn't see how to obtain the wave-function from there. So this doesn't seem to be as easy as I thought. I was actually hoping that it's just a standard textbook problem. $\endgroup$
    – André
    Commented Aug 11, 2015 at 21:15
  • $\begingroup$ If all you need is a rough order of magnitude estimate there are simple ways to do it. Approximate the interparticle potential as harmonic so that equipartition theorem holds. So then the average potential energy per particle is like <U> = (1/2)k <(Delta x)^2> = k_B T/2, where k_B is Boltzmann's constant, T is temp. in Kelvin, Delta x is particle distance from equilibrium, k is an effective spring constant. So this let's you find a typical deviation from equilibrium. The effective spring constant for a Si-Si bond should be tabulated somewhere. $\endgroup$
    – gleedadswell
    Commented Aug 12, 2015 at 20:12
  • $\begingroup$ gleedadswell, thanks, but as far as I understand, the approximation by single harmonic oscillators is exactly Einstein's model, which breaks down for low temperatures? As far as I could find out so far, the T proportionality is valid down to ~40-50 K, and below sqrt(<x^2>) goes to a constant value around 5*10^-12 m for Si. I found some nice papers with theoretical predictions and experimental results from neutron and x-ray scattering. With this I know at least the rough size. I couldn't find a result for the right shape of the mass distribution inside the crystal yet. $\endgroup$
    – André
    Commented Aug 16, 2015 at 18:08

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