Is there anyway to use a scientific instrument to measure the density of electron around the atomic orbital? Is there anyway to use a scientific instrument to measure the density of electron around the atomic orbital? Please list both old way and more modern ways.
 A: Old ways used Schrodinger's equation's solutions for the atoms and mapped the square of the wave function.. Since the solution fitted the spectrum of the atom it was accepted that the orbital was also correct.
Recently there has been an experiment that measured the orbitals of the hydrogen atom
The abstract from the link:

To describe the microscopic properties of matter, quantum mechanics uses wave functions, whose structure and time dependence is governed by the Schrödinger equation. In atoms the charge distributions described by the wave function are rarely observed. The hydrogen atom is unique, since it only has one electron and, in a dc electric field, the Stark Hamiltonian is exactly separable in terms of parabolic coordinates (η, ξ, φ). As a result, the microscopic wave function along the ξ coordinate that exists in the vicinity of the atom, and the projection of the continuum wave function measured at a macroscopic distance, share the same nodal structure. In this Letter, we report photoionization microscopy experiments where this nodal structure is directly observed. The experiments provide a validation of theoretical predictions that have been made over the last three decades.

. A popularization is here.


After zapping the atom with laser pulses, ionized electrons escaped and followed a particular trajectory to a 2D detector (a dual microchannel plate [MCP] detector placed perpendicular to the field itself). There are many trajectories that can be taken by the electrons to reach the same point on the detector, thus providing the researchers with a set of interference patterns — patterns that reflected the nodal structure of the wave function.
And the researchers managed to do so by using an electrostatic lens that magnified the outgoing electron wave more than 20,000 times.

Please note that the orbitals are a probability distribution for finding an electron in a specific (x,y,z) around the nucleus, not a matter density in the classical sense. This experiment is for one electron and from the description it does not seem it would work for higher atomic numbers, at least not as simply.
A: The older method would be using scattering. If you shoot an electron at an atom, it will get scattered by the electrons. The exact shape of the repelling potential (that depends on the density distribution), conditions the outgoing angles.
This is, in principle, valid for polyelectronic atoms, but deconvolving the output to infer a potential is a quite complicated problem, and it gets very hard if the number of electrons is high. Not to mention the very wavefunction for two electrons is already impossible to get analytically.
